How to prove that a sequence with recurrence relation has no closed form expression? It is always easy to forge recurrence relations. E.g.
$$a_{n+1}=2a_n+\dfrac{1}{a_n}, a_0=1$$
But it is always hard to find the general closed form expression. And it is even harder to prove that there is no such expression.
Another example is here. And I highly suspect sequences like
$$a_n = \sum_{i = 0}^n 10^{-2^i}$$ have a closed form expression.
Is it possible to prove that a closed form expression does not exist in the above cases? Are there any theories on this kind of problem?

A general theory of impossibility proofs (TOIP) would be the holy grail.
Note that impossibility proofs are the hardest of all: You have to prove that all conceivable constructions of a certain theory (say, constructions with ruler and compass, or "formulas in terms of elementary functions") cannot solve a certain problem you are interested in. For such a proof you need a theory that transcends the scope of your constructions; e.g., Galois theory for the problem of trisecting an angle.
Concerning the problem at hand: There is a theory that allows you to decide whether some given  elementary function has an elementary primitive. But this theory (which includes an impossibility proof for many important cases) is very deep, and 99.9% of students never see a proof that $x\mapsto e^{-x^2/2}$ has no elementary primitive. There is no such theory for functions $n\mapsto a_n$ defined recursively by a formula $a_{n+1}=\Phi(a_n)$, or similar, where $\Phi(\cdot)$ is some elementary expression.
