I dreamt of a data structure, does it exist?
You're describing the classical Bentley-Saxe logarithmic method, applied to static sorted arrays.  The same idea can be used to add support for insertions to any static data structure (no insertions or deletions) for any decomposable searching problem.  (A search problem is decomposable if the answer for any union $A\cup B$ can be computed easily from the answers for the sets $A$ and $B$.)  The transformation increases the amortized query time by a factor of $O(\log n)$ (unless it was already bigger than some polynomial in $n$), but increases the space by only a constant factor.  Yes, it can be deamortized, thanks to Overmars and van Leeuwen, but you really don't want to do that if you don't have to.

These notes cover the basics.

Cache-oblivious lookahead arrays are the mutant offspring of Bentley-Saxe and van Emde Boas trees on steroids.