Here is a small discovery I stumbled across a few weeks ago. I hope at least one person will find it interesting enough to help me. The iterated continued fractions from convergents (or convergents constants) are explained at https://oeis.org/wiki/Convergents_constant . Among other things, there you find out that a randomly selected number between 2 and 3 has a convergents constant of 2.3484074702792306..., but the same is not true for 2.1, 2.2, 2.5 and perhaps for a few other values. Also for most 0 < x < 1 the same process of iterated continued fractions returns 0.5557531042780459..., but it is not so for x =0.1, 0.11, 0.12, 0.2, 0.25, 0.34, 0.35, 0.43, 0.45, 049, 0.5, 0.65, 0.75, some values < 66/1477 and perhaps for a few other values.
I would like to know why these numbers and those like them are exceptions to the rule.