Close: my answer would have been the first index is Φ ≈ 1.618, second is e ≈ 2.718, third is π ≈ 3.14, and so on, picking irrational (and preferably transcendental) numbers whose integer portion is the next counting number. (I had forgotten that Φ, like √2, is irrational but algebraic, and had started to write that as "picking transcendental numbers" before having a long think about it; still, I prefer Φ since it has a predefined symbol.)

Ooh, another fun set: ln(2) ≈ 0.693, √2 ≈ 1.414, 2**(√2) ≈ 2.665, ... I'm sure someone could come up with a whole series of those whose integer portions are increasing, but all the terms only combine operators and 2. ;-)