Oh boy, simple answer is a whole bunch. I have boxes of flies to fish from trout to salmon, steelhead, tarpon, snook, bonefish, smallmouth, etc.... My favorite are some of the C&F boxes and the old school Myran clear plastic boxes. I own a couple Wheatly boxes, theyre nice but not the only good choice out there. Basically, I look for simple features, quality closure and reasonable price. Many of my big fly boxes are standard bass fishing boxes that you might use for jerk baits. These really work great for the big 8,9,10wt flies when I'm in a boat. My biggest recommendation for trout flys is to keep it simple and non distructive on your flys. As examples-'those Wheatly boxes with all those little hinged lids are cool but a pain and many boxes can smash down hackles on drys.
I carry 3 boxes currently:
1. small, light foam box - caddis. If you want light and functional, I recommend these boxes, I carried the same for about 8 years, that brought the price down to...I don't know, basically free for the number of days used. They have magnets on the corners, float and are weightless.
2. small, two-sided, slit foam box with clear plastic lids from SA. I like these boxes a lot, too. This little box actually has three "pages" for flies, the middle sleeve plus the lid of one side that I fill with wet flies. This is my mayfly box.
3. a large CF nymph box, one side with compartments one side with slit foam rows. This is also a very versatile box. The compartments carry large dries i.e. hoppers, stimis, ect. plus small streamers and big nymphs. The slit side carries nymphs. These are expensive though at ~$50.
Forgot 4. a soft streamer wallet that disappears anywhere in the pack.
I think the best boxes for the money are the Scientific Angler water proof line. I haven't gone so far as the "X System" of interchangeable pages (seems like gimmick). Slit foam took a long time to get here, and I'm glad it did. They are plastic, light, the float and seal shut. They are a very functional, inexpensive version of the C&F boxes. And they come in a variety of sizes and configurations and are color coded. Check em out.
I highly recommend them after searching for the ideal box for a long time.
I have two main boxes in my vest, dry flys and wet/nymphs, oh then there are the other two little ones for the tiny midges and smaller drys, oh then there is one at the tying desk and then there are the ones in the shed of years gone by, back ups ya know you can never have enough back ups, then there are the others in the.....well you get the idea.
I have quite a few, but all I really use now are the SA boxes. One for nymphs & wet flies, and one for streamers. I don't fish dries very often, but usually they are really small flies that fit in the nymph box.
I do think I will look at one of the really thin boxes for nymphs though.
Apparently you can never have too many fly boxes......