I have been fly fishing for many years and I generally fly fishing with a 2-fly rig...ie. I have 2 flies on the same line. For many reasons people fish like this which essentially doubles your odds of catching a fish, but there are many other reasons why to fish with 2 flies.
To answer your question, if I was forced to choose 2 flies for any river in the world to fish with for the rest of my life I would choose a Red Copper John and a Barr emerger. These 2 flies will work almost anywhere,and any time of year in almost any circumstance. If I was given the chance to add a 3rd fly to my arsenal, it would be a tough choice between a Pheasant Tail and an Elk Hair Caddis, but probably the Caddis.
Thats just me, but if you don't have those 2 flies and at least one of the other 2 you won't catch as many trout than if you did have them.