This list is meant to be suggestions. It is not comprehensive. Feeding finches (or any pet bird) is something that needs a little thought and a little preparation. These fruits and vegetables can be chopped small or left in larger chunks.
Apple, no seeds or skin
Cabbage, both savoy and kale
Corn on or off the cob
Nasturtium flowers and leaves
Tomatoes, in small amounts and ripe
This list is as useful for any size of bird, except those with specific dietary considerations. Please also check out Foods you should Never give your Bird.