Can one describe the set $\{e^f+e^g: f, g\in H(C)\}$ in some way? For example, in unital Banach algebras, every element has this form. I am in particular interested in the problem whether the function $u=(1-e^L)^2$ with $L=(1-z)^{-3}$ is a sum of two exponentials in H(D). Here $L$ is chosen so that every value is taken by u in D (hopefully).

Describing the set $\{ e^f+e^g:f,g\in H(C)\}$ is difficult, and probably it cannot be described in any reasonable form. However many conditions can be given for a function $h$ not to be of this form. For example, $h$ cannot be a square (or any power) of an entire function. Proof: Suppose that there are entire functions $f,g,w$ such that $e^f+e^g=w^n$, and $w$ is not constant. Then $e^{f-g}$ omits zero, and all solutions of $e^{f(z)-g(z)}+1=0$ are multiple. By the Second Main theorem of Nevanlinna, $f-g$ must be constant, so $w$ must be constant, a contradiction.

For functions in the unit disk, this argument will prove that functions of sufficiently fast growth are not representable in the form $e^f+e^g$. In particular, this covers your example. Indeed, using the notation of Nevanlinna theory, we have $$T(r,u)\geq m(r,u)\geq c(1-r)^{-2},$$ so the conditions of the Second Main Theorem hold, and the above argument shows that $u$ is not a sum of two exponentials.

Reference: Hayman, Meromorphic functions, Chap. 2.