I tie a lot of flies more as a hobby than anything and love to fish the flies I tie.
However, I honestly think I could catch just as many fish with a very few flies:
Adams - 12, 14, 16, 18
Bead Head Pheasant Tail - same sizes
Bead Head Hares Ear - same sizes
On tail waters add
Griffith Ganat - size 18 and 20
For warm water:
Black wooly bugger - size 8, 10, 12
White wooly bugger - same sizes
foam popper - small, med, large yellow with yellow tail
bead head pheasant tail - size 12 and 14
Marabou Steamer - yellow.
I honestly think with the above flies, one can catch just about any fresh water fish worth catching on a fly rod.
Now, if you look in my fly box, you're oing to find all of these and a whole bunch of other flies too.