deforming geometric series

Yesterday I asked if there could be such a sequence of real numbers \{a_{n}\}_{\infty} such that for all n\in\mathbb{N}, AGM(a_{n},a_{n+1}) = 1/2^{n}, and it turns out that the new sequence is just a linearly stretched version, and that had me wondering, in what cases are we aware of nonlinearly stretched geometric sequences:

The ordinary geometric sequence:
\sum_{n=1}^{\infty} \frac{1}{2^{n}} = 1

The exponential:
\sum_{n=1}^{\infty} \frac{1}{2^{n}n!} = \sqrt{e}-1

“Wahi” or “Upsilon” functions:
\sum_{n=1}^{\infty} \frac{1}{2^{n}n^{n}} = \int_{0}^{1} \frac{dx}{2\sqrt{x^{x}}} = \Upsilon_{0}(1/2) = ʍ(1/2)-1

The polylogarithm (in particular, the dilogarithm):
\sum_{n=1}^{\infty} \frac{1}{2^{n}n^{2}} = Li_2(1/2)

But there are lacunary monsters like the following:
\sum_{n=1}^{\infty} \frac{1}{2^{n}3^{2^{n}}}
That there are no present methods of evaluation for.

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