Thanks for the many excellent suggestions to think about! Some of them might work; I'll have to ponder further. Just to be clear, I'm asking for a bit MORE than just an existence proof that gives no hint of how to find an example. There are indeed a number of notorious cases of those (I appreciate the reminders). However what I want is one of those, PLUS an actual non-trivial example case that's reasonably easily verifiable. The example shouldn't be immediately obvious, and would probably be obtained using either more advanced methods, or else just brute force. As an illustrative analogy: an elementary non-constructive proof that there are integers that are sums of two cubes in two different ways, PLUS a statement like, "oh, and by the way, 1729 is the smallest such, which you can easily verify by 12^3+1^3=10^3+9^3" or the like would hit the spot. (The second "1729" part of this fills the bill, but the first part is either obvious, or else could entail showing that some generating function wasn't identically zero, which isn't exactly "elementary"). Perhaps Gene's suggestion can be made to work. Certainly the wonderful Cantor diagonal proof is easy enough. But traditionally we'd show the transcendence of some particular example Liouville number X in two stages, by first proving that ALL Liouville numbers are transcendental, and then showing that X is Liouville. Can we keep this argument sufficiently elementary yet make it more direct? Thanks!