There's a very important distinction to be made when discussing wheat allergies: Is the person you are feeding simply allergic to wheat, or do they have a full-fledged gluten intolerance?
If the latter case is true, I refer you to the many excellent resources at Gluten free diet and Gluten free recipes. For you whose immune system's only quarrel is with wheat in particular, I have a small announcement:
You are not doomed to a life of dry, crumbly bread!
My father is allergic to wheat, potatoes, milk, and most varieties of nuts. While I have not (yet!) been cursed with his handicap, my family cooks to accomodate his needs — and we're vegetarians to boot, so I'm conversant in a mode of cookery that most Americans would call a food-free diet. On this authority, I reveal to you the great secret of wheat-free-but-having-a-bit-of-gluten cookery:
Barley. Spelt. Rye.
Barley is my favorite all-around flour for most kinds of baking that don't need a high gluten content. Anything other than bread, in other words. Barley has a nice smooth texture and a little bit of a sweet flavor to begin with (I've often heard the taste described as "cakelike"), so it's a natural for cakes. I did a side-by-side test recently, making two "identical" chocolate cakes — one whole wheat, one barley. We all preferred the barley cake; It held the line on "moist" without degenerating into pudding, even in the face of an onslaught of honey and cocoa that left the wheat cake slightly on the too-gooey side. Barley is great for cookies, muffins and pancakes, too.
If you're baking bread, I recommend you try spelt. It's a lot like wheat (In fact, it's a recent ancestor. Spelt is often referred to as a
primitive wheat), but it's different enough that some wheat-sensitive folks can eat it safely. If I want to allergy-proof a recipe without thinking too much, I replace wheat with spelt, unit for unit.1 The flavor is fairly wheatlike, perhaps a touch richer and slightly more prone to bitterness if you undercook it. The main disadvantage of spelt is the price — spelt flour seems to command double the price of whole wheat. My mother, intrepid gardener that she is, grew some and discovered why. It seems that spelt has a husk that is harder to thresh off (several orders of magnitude harder) than the husk on modern varieties of wheat.
Rye isn't quite as versatile as other grains, if only by virtue of its more intense flavor—I, personally, don't have much interest in a rye-flavored pie crust. Rye is more of a specialist: Give it a job it's good at and stand back. Rye is where it's at for heavy sourdough breads, and vice versa. Rye doughs tend to be dense and very sticky, especially when they're warm, making a full-bodied bread that you'll want to chew long and well, with a flavor worth lingering over. Please don't confuse the flavor of rye with the sorely-abused flavors of caraway and molasses; most commercial "rye" breads are mostly wheat and contain as much caraway as they do rye flour. Presumably there is a massive conspiracy afoot (sponsored by the National Caraway Council, working in tandem with the Wheat Supremacy Board) to dupe the bread eaters of the US into settling for slighty doped Wunder Bredd.
Rye kernels, and by extension freshly ground rye flour that hasn't been killed by heat in a high-speed grinder, tend to have an abundance of natural yeasts on them, so it's one of the easier grains to use if you want to create a sourdough starter. A starter you make this way will be rye-optimized—sourdoughs, like presidents, need some time to settle in before they can work well with the grain (or policies, in the other half of the simile) you feed them. You should also remember that a sourdough contains some of whatever flour you feed it, so a wheat-based sourdough will put a small amount of wheat flour into your bread. The concentration will drop in a sort of diminishing inverse-square curve, diluting more every batch until only the homeopathic memory of it remains. If you're extremely sensitive to wheat, you may want to simply start your own sourdough in rye or else let someone else eat the first five or ten batches of your bread.
That Other Stuff
Rice is great stuff, but it just doesn't cut it as a bread flour. It's entirely too crumbly, and it also tends to make a loaf of throat-dessicating dryness when used in pure form. Boil it whole or save the flour for your gluten-intolerant friends. The same goes for potatoes, tapioca, millet and any kind of bean: eat them, enjoy them, grind them into flour if you must, but please do not pretend that they are all-purpose wheat flour substitutes.
1I ought to reveal that I never follow recipes very closely, so there may be — probably are — some minor adjusments involved in changing grains which I just compensate for unconsciously. You'll be doing the same as soon as you get accustomed to the differences, so don't sweat it.