Liouville numbers such as $$\sum_{k=1}^\infty\frac1{10^{k!}}$$ are known to be transcendental, essentially from Diophantine approximation type arguments. Using stronger results than what Louiville had immediate access to (like Roth's theorem), I think we can show numbers such as $$\sum_{k=1}^\infty\frac1{10^{3^k}}$$ are transcendental as well, but I don't know how to do better than this.
As an actual question, is it known if $$\sum_{k=1}^\infty\frac1{10^{k^2}}$$ is transcendental? What is the state of the art here?