diff --git a/django/contrib/gis/db/backends/base/operations.py b/django/contrib/gis/db/backends/base/operations.py
index dc2fad025b..4759c86b89 100644
--- a/django/contrib/gis/db/backends/base/operations.py
+++ b/django/contrib/gis/db/backends/base/operations.py
@@ -98,12 +98,12 @@ class BaseSpatialOperations(object):
         """
         raise NotImplementedError('subclasses of BaseSpatialOperations must provide a geo_db_placeholder() method')
 
-    def check_aggregate_support(self, aggregate):
-        if isinstance(aggregate, self.disallowed_aggregates):
+    def check_expression_support(self, expression):
+        if isinstance(expression, self.disallowed_aggregates):
             raise NotImplementedError(
-                "%s spatial aggregation is not supported by this database backend." % aggregate.name
+                "%s spatial aggregation is not supported by this database backend." % expression.name
             )
-        super(BaseSpatialOperations, self).check_aggregate_support(aggregate)
+        super(BaseSpatialOperations, self).check_expression_support(expression)
 
     def spatial_aggregate_name(self, agg_name):
         raise NotImplementedError('Aggregate support not implemented for this spatial backend.')
diff --git a/django/contrib/gis/db/models/aggregates.py b/django/contrib/gis/db/models/aggregates.py
index 0ec842de0f..42198d9287 100644
--- a/django/contrib/gis/db/models/aggregates.py
+++ b/django/contrib/gis/db/models/aggregates.py
@@ -9,6 +9,9 @@ class GeoAggregate(Aggregate):
     is_extent = False
 
     def as_sql(self, compiler, connection):
+        # this will be called again in parent, but it's needed now - before
+        # we get the spatial_aggregate_name
+        connection.ops.check_expression_support(self)
         self.function = connection.ops.spatial_aggregate_name(self.name)
         return super(GeoAggregate, self).as_sql(compiler, connection)
 
diff --git a/django/db/backends/base/features.py b/django/db/backends/base/features.py
index 0f6ee0efe3..4b4d5c6d75 100644
--- a/django/db/backends/base/features.py
+++ b/django/db/backends/base/features.py
@@ -1,3 +1,5 @@
+from django.db.models.aggregates import StdDev
+from django.db.models.expressions import Value
 from django.db.utils import ProgrammingError
 from django.utils.functional import cached_property
 
@@ -226,12 +228,8 @@ class BaseDatabaseFeatures(object):
     @cached_property
     def supports_stddev(self):
         """Confirm support for STDDEV and related stats functions."""
-        class StdDevPop(object):
-            contains_aggregate = True
-            sql_function = 'STDDEV_POP'
-
         try:
-            self.connection.ops.check_aggregate_support(StdDevPop())
+            self.connection.ops.check_expression_support(StdDev(Value(1)))
             return True
         except NotImplementedError:
             return False
diff --git a/django/db/backends/base/operations.py b/django/db/backends/base/operations.py
index 24bcbb3d08..f535a8792d 100644
--- a/django/db/backends/base/operations.py
+++ b/django/db/backends/base/operations.py
@@ -526,12 +526,16 @@ class BaseDatabaseOperations(object):
         return value
 
     def check_aggregate_support(self, aggregate_func):
-        """Check that the backend supports the provided aggregate
+        return self.check_expression_support(aggregate_func)
 
-        This is used on specific backends to rule out known aggregates
-        that are known to have faulty implementations. If the named
-        aggregate function has a known problem, the backend should
-        raise NotImplementedError.
+    def check_expression_support(self, expression):
+        """
+        Check that the backend supports the provided expression.
+
+        This is used on specific backends to rule out known expressions
+        that have problematic or nonexistent implementations. If the
+        expression has a known problem, the backend should raise
+        NotImplementedError.
         """
         pass
 
diff --git a/django/db/backends/sqlite3/features.py b/django/db/backends/sqlite3/features.py
index fa5a002603..ee86469177 100644
--- a/django/db/backends/sqlite3/features.py
+++ b/django/db/backends/sqlite3/features.py
@@ -60,7 +60,7 @@ class DatabaseFeatures(BaseDatabaseFeatures):
         """Confirm support for STDDEV and related stats functions
 
         SQLite supports STDDEV as an extension package; so
-        connection.ops.check_aggregate_support() can't unilaterally
+        connection.ops.check_expression_support() can't unilaterally
         rule out support for STDDEV. We need to manually check
         whether the call works.
         """
diff --git a/django/db/backends/sqlite3/operations.py b/django/db/backends/sqlite3/operations.py
index ad05f88585..cd29092cd7 100644
--- a/django/db/backends/sqlite3/operations.py
+++ b/django/db/backends/sqlite3/operations.py
@@ -4,7 +4,7 @@ import datetime
 import uuid
 
 from django.conf import settings
-from django.core.exceptions import ImproperlyConfigured
+from django.core.exceptions import ImproperlyConfigured, FieldError
 from django.db import utils
 from django.db.backends import utils as backend_utils
 from django.db.backends.base.operations import BaseDatabaseOperations
@@ -33,15 +33,21 @@ class DatabaseOperations(BaseDatabaseOperations):
         limit = 999 if len(fields) > 1 else 500
         return (limit // len(fields)) if len(fields) > 0 else len(objs)
 
-    def check_aggregate_support(self, aggregate):
+    def check_expression_support(self, expression):
         bad_fields = (fields.DateField, fields.DateTimeField, fields.TimeField)
-        bad_aggregates = (aggregates.Sum, aggregates.Avg,
-                          aggregates.Variance, aggregates.StdDev)
-        if aggregate.refs_field(bad_aggregates, bad_fields):
-            raise NotImplementedError(
-                'You cannot use Sum, Avg, StdDev and Variance aggregations '
-                'on date/time fields in sqlite3 '
-                'since date/time is saved as text.')
+        bad_aggregates = (aggregates.Sum, aggregates.Avg, aggregates.Variance, aggregates.StdDev)
+        if isinstance(expression, bad_aggregates):
+            try:
+                output_field = expression.input_field.output_field
+                if isinstance(output_field, bad_fields):
+                    raise NotImplementedError(
+                        'You cannot use Sum, Avg, StdDev and Variance aggregations '
+                        'on date/time fields in sqlite3 '
+                        'since date/time is saved as text.')
+            except FieldError:
+                # not every sub-expression has an output_field which is fine to
+                # ignore
+                pass
 
     def date_extract_sql(self, lookup_type, field_name):
         # sqlite doesn't support extract, so we fake it with the user-defined
diff --git a/django/db/models/aggregates.py b/django/db/models/aggregates.py
index 06220123ca..668f79f622 100644
--- a/django/db/models/aggregates.py
+++ b/django/db/models/aggregates.py
@@ -25,17 +25,6 @@ class Aggregate(Func):
         c._patch_aggregate(query)  # backward-compatibility support
         return c
 
-    def refs_field(self, aggregate_types, field_types):
-        try:
-            return (isinstance(self, aggregate_types) and
-                    isinstance(self.input_field._output_field_or_none, field_types))
-        except FieldError:
-            # Sometimes we don't know the input_field's output type (for example,
-            # doing Sum(F('datetimefield') + F('datefield'), output_type=DateTimeField())
-            # is OK, but the Expression(F('datetimefield') + F('datefield')) doesn't
-            # have any output field.
-            return False
-
     @property
     def input_field(self):
         return self.source_expressions[0]
diff --git a/django/db/models/expressions.py b/django/db/models/expressions.py
index 97a2a9071d..fb094fd4ef 100644
--- a/django/db/models/expressions.py
+++ b/django/db/models/expressions.py
@@ -297,14 +297,6 @@ class BaseExpression(object):
                 return agg, lookup
         return False, ()
 
-    def refs_field(self, aggregate_types, field_types):
-        """
-        Helper method for check_aggregate_support on backends
-        """
-        return any(
-            node.refs_field(aggregate_types, field_types)
-            for node in self.get_source_expressions())
-
     def prepare_database_save(self, field):
         return self
 
@@ -401,6 +393,7 @@ class DurationExpression(Expression):
         return compiler.compile(side)
 
     def as_sql(self, compiler, connection):
+        connection.ops.check_expression_support(self)
         expressions = []
         expression_params = []
         sql, params = self.compile(self.lhs, compiler, connection)
@@ -473,6 +466,7 @@ class Func(ExpressionNode):
         return c
 
     def as_sql(self, compiler, connection, function=None, template=None):
+        connection.ops.check_expression_support(self)
         sql_parts = []
         params = []
         for arg in self.source_expressions:
@@ -511,6 +505,7 @@ class Value(ExpressionNode):
         self.value = value
 
     def as_sql(self, compiler, connection):
+        connection.ops.check_expression_support(self)
         val = self.value
         # check _output_field to avoid triggering an exception
         if self._output_field is not None:
@@ -536,6 +531,7 @@ class Value(ExpressionNode):
 
 class DurationValue(Value):
     def as_sql(self, compiler, connection):
+        connection.ops.check_expression_support(self)
         if (connection.features.has_native_duration_field and
                 connection.features.driver_supports_timedelta_args):
             return super(DurationValue, self).as_sql(compiler, connection)
@@ -650,6 +646,7 @@ class When(ExpressionNode):
         return c
 
     def as_sql(self, compiler, connection, template=None):
+        connection.ops.check_expression_support(self)
         template_params = {}
         sql_params = []
         condition_sql, condition_params = compiler.compile(self.condition)
@@ -715,6 +712,7 @@ class Case(ExpressionNode):
         return c
 
     def as_sql(self, compiler, connection, template=None, extra=None):
+        connection.ops.check_expression_support(self)
         if not self.cases:
             return compiler.compile(self.default)
         template_params = dict(extra) if extra else {}
@@ -851,6 +849,7 @@ class OrderBy(BaseExpression):
         return [self.expression]
 
     def as_sql(self, compiler, connection):
+        connection.ops.check_expression_support(self)
         expression_sql, params = compiler.compile(self.expression)
         placeholders = {'expression': expression_sql}
         placeholders['ordering'] = 'DESC' if self.descending else 'ASC'
diff --git a/django/db/models/sql/query.py b/django/db/models/sql/query.py
index 9693206b67..0d89de2458 100644
--- a/django/db/models/sql/query.py
+++ b/django/db/models/sql/query.py
@@ -230,11 +230,6 @@ class Query(object):
             raise ValueError("Need either using or connection")
         if using:
             connection = connections[using]
-
-        # Check that the compiler will be able to execute the query
-        for alias, annotation in self.annotation_select.items():
-            connection.ops.check_aggregate_support(annotation)
-
         return connection.ops.compiler(self.compiler)(self, connection, using)
 
     def get_meta(self):
diff --git a/django/db/models/sql/where.py b/django/db/models/sql/where.py
index e3766c51d6..6a03210a93 100644
--- a/django/db/models/sql/where.py
+++ b/django/db/models/sql/where.py
@@ -325,17 +325,6 @@ class WhereNode(tree.Node):
     def contains_aggregate(self):
         return self._contains_aggregate(self)
 
-    @classmethod
-    def _refs_field(cls, obj, aggregate_types, field_types):
-        if not isinstance(obj, tree.Node):
-            if hasattr(obj.rhs, 'refs_field'):
-                return obj.rhs.refs_field(aggregate_types, field_types)
-            return False
-        return any(cls._refs_field(c, aggregate_types, field_types) for c in obj.children)
-
-    def refs_field(self, aggregate_types, field_types):
-        return self._refs_field(self, aggregate_types, field_types)
-
 
 class EmptyWhere(WhereNode):
     def add(self, data, connector):
diff --git a/tests/backends/tests.py b/tests/backends/tests.py
index 4a07cd5f2f..979ebefa01 100644
--- a/tests/backends/tests.py
+++ b/tests/backends/tests.py
@@ -128,12 +128,19 @@ class SQLiteTests(TestCase):
         #19360: Raise NotImplementedError when aggregating on date/time fields.
         """
         for aggregate in (Sum, Avg, Variance, StdDev):
-            self.assertRaises(NotImplementedError,
+            self.assertRaises(
+                NotImplementedError,
                 models.Item.objects.all().aggregate, aggregate('time'))
-            self.assertRaises(NotImplementedError,
+            self.assertRaises(
+                NotImplementedError,
                 models.Item.objects.all().aggregate, aggregate('date'))
-            self.assertRaises(NotImplementedError,
+            self.assertRaises(
+                NotImplementedError,
                 models.Item.objects.all().aggregate, aggregate('last_modified'))
+            self.assertRaises(
+                NotImplementedError,
+                models.Item.objects.all().aggregate,
+                **{'complex': aggregate('last_modified') + aggregate('last_modified')})
 
 
 @unittest.skipUnless(connection.vendor == 'postgresql', "Test only for PostgreSQL")