The counting in "first, second, third" quantization only has a mathematical meaning - it quantifies a hierarchy of numbers, functions, functionals, functionalals, and so on that appear in the most straightforward description of the classical configuration at time $t$; or the wave function at time $t$.
Classical mechanics is "zeroth quantization" and it deals with numbers like $x,p$ - classical positions and momenta - at a given time. Quantum mechanics is obtained by promoting $x,p$ to operators, then $\psi(x)$, a function, expresses the state at a given moment.
Formally speaking, quantum field theory is approximately a result of the second quantization in which $\psi(x)$ is promoted to operators again - approximately field operators $\Phi(x,y,z)$ - and the state vector is given via functionals $\Psi[\Phi(x,y,z)]$. The number of numbers in this functional is "hierarchically higher" than it is for regular functions.
In string field theory, one formally goes one step further. The string fields $\Psi[x(\sigma)]$ which are themselves functionals, even in the classical (non-quantum) string field theory, are promoted to operators, so the state vector may be written as $\bar\Psi \{ \Psi[x(\sigma)] \} $, a functionalal.
In principle, you could imagine theories going even higher, to more complicated constructs. However, even the term "second quantization" is problematic because the wave function may also be written differently, without a functional - e.g. as a collection of functions of finitely many variables (wave functions for $n$ particles). That's just a result of a difference choice of a basis.
All physically relevant (which also includes the condition "separable") infinite-dimensional Hilbert spaces in quantum mechanical theories are isomorphic to each other, regardless whether they are obtained by third, second, or just first quantization. So all quantizations are physically really the same thing, qualitatively speaking, and the counting is just describing a heuristic way to quantify the number of numbers in the state vector that is however basis-dependent.
That's why even the "second quantization" term is outdated. Mathematically speaking, the QFT state vector may be written as a functional. But a QFT itself is a (first) quantization of a classical system, namely a classical field theory. It is not "another quantization" of the state vector $\psi(x,y,z)$ in non-relativistic quantum mechanics because this wave function is physically different from a classical field - they have very different interpretations even though, in some sense, they contain approximately "the same number of numbers".
To consider the fourth, fifth, or infinity-th quantization may be appealing but this identification is basis-dependent and one would still need some physical principles to pinpoint the interesting theories of the kind, so this method of getting more advanced theories probably isn't sufficient to do genuine progress in physics.
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