We need a way to update a value by increasing it in a way that it converges. However, it should be done without keeping a counter of the number of operations, so $n$ should not be involved in the calculations. We tried
$$ a_0 = 0,\quad a_1 = 1,\quad a_{n+1} = a_n + \frac{1}{a_n} $$
but it does not converge. So we thought about the golden ratio sequence ($a_{n+1} = 1 + 1/a_n$) but it oscillates. Now we are thinking about
$$ a_0 = 0,\quad a_n = a_{n-1}+\frac{1}{e^{a_{n-1}}}. $$
It looks promising, but does it converge? Any suggestions of alternative sequences?