A summation series

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I am asking on the spur, so there has not been too much thought put into it, but how would we classify a series summation such as $$ \sum_{i=0}^{n} 2^{2^i} ~ ?$$ It does not feel to be geometric, nor that it can be made to be geometric. In general, the function xx does not look like it bears a Taylor expansion, so I don't think it even has an algebraic approximation (in which case, I doubt it could be the root of a power series either).

I was considering the case |q| < 1, where if we took an infinite product of a summation $$ \sum_{i=0}^{n} q^{2^i} ~ ?$$ in some particular way, we might have a convergent power series (or vice versa). Is there already literature on this sort of thing?
 
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StoneTemplePython
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Your question seems to, curiously, be related to the problem of the week.

https://www.physicsforums.com/threads/intermediate-math-problem-of-the-week-12-11-2017.934137/

specifically, if you take the log of the infinite product to convert to infinite series, recognize you're in radius of convergence for natural log, expand the series (giving you ##\sum\sum##), and upper bound ##\frac{1}{
2^{2^n}+2^{-2^n}} \leq
\frac{1}{
2^{2^n}}## and then interchange the summations...
 

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