Are there versions of String Theory formulated in $D$ spacetime dimensions or even in infinitely many dimensions? There are a lot of different versions of string theory, and almost all of them differ in the number of dimensions. The most famous ones are formulated in 10, 11 or 26 dimensions.
But are there any versions of string theory that are formulated not in a fixed number of dimensions, but which are consistent in any number or even in an infinite number of them?
In your article "Superstrings: A Theory of Everything?" you say at a certain point that:
"Talking about four or ten dimensions at all is itself only an approximation to this much larger stringy space which really has an infinite number of dimensions"
Edit: In Paul C Davies book "Superstrings: A Theory of Everything?", it includes a discussion section with one of the founders of String Theory, Michael B Green, who says at a certain point that:
"Talking about four or ten dimensions at all is itself only an approximation to this much larger stringy space which really has an infinite number of dimensions"
Does this mean that String Theory is actually in infinitely many dimensions?
Than you in advance for your help
 A: I'd like to point out first that the $d=26$ theory I'm assuming is bosonic string theory, and that's not one we hold to be a candidate for anything in reality, because it excludes fermions. It's used more as a toy model.
However, let's stick with it for a moment. In principle, you could pick any $d$ for bosonic string theory, however the only choice preserving $SO(1,d-1)$ symmetry is for the choice $d=26$.$^\dagger$
Similarly, other string theories in 10 dimensions are in 10 dimensions because of consistency requirements. If you didn't care about consistency, there are other possibilities.
As for the 11 dimensional theory you speak of, this refers to the fact M-theory is described by 11 dimensional supergravity, but this isn't the same as a string theory. Rather, it is a theory wherein particular limits correspond to the known string theories.

$\dagger$ An alternative argument is that the Weyl symmetry is anomalous unless $c=0$, which we can only do by adding $26$ scalar fields (to the $c=-26$ ghost system), which in turn fixes $d=26$, as each $X^i$ is basically a scalar field from the target space perspective. 
A: Yes. It is possible to formulate string theory in D dimensions :) Recall that the preferred dimension for string theory is taken to be D=10 (or D=11 in M-theory, D=26 in bosonic string theory, or D=4 in N=2,4 strings) because that is the only dimension in which is possible to consistently quantize propagating strings with a Lorentz/Weyl-invariant spectrum; if you relax that requirement, any dimension is alloweed. Even string theories with multiple (or even emergent) dimensions of time are allowed
References:
Critical dimension: anything goes?
de Sitter Space in Non-Critical String Theory
Charting the landscape of supercritical string theory
