Are numbers such as this one non-algebraic (transcendental)?
The number starts: 0.05002500012500000625000...
It has an endless series of numbers, each term of which appears at ever increasing spacing in the digits.
I've used an infinite sum, algebraic numbers need to be able to be represented with a finite number of operations and cannot use infinite operations such as sums and products. However there may be an equivalent formula that uses only whats aloud for an algebraic number.
The full formula/number:
https://www.wolframalpha.com/input/?i=sum+1%2F((2%5En)*(10%5E(n%5E2))),+n%3D1+to+infinity
Your number is $$ \sum_{n=1}^\infty 2^{-n} 10^{-n^2}$$ It's almost certainly transcendental, but I don't know if it's possible to prove that in the current state of the art. Proving transcendence is notoriously difficult. I might note that $\sum_{n=1}^\infty 10^{-n^2}$, which is essentially a Jacobi theta function evaluated at $1/10$, is known to be transcendental: see e.g. this paper by Daniel Duverney, Keiji Nishioka, Kumiko Nishioka, and Iekata Shiokawa. It may be that the techniques of that paper could be applied to this problem: I haven't tried.