The Fundamental theorem of arithmetic gives a natural way to write natural numbers $\mathbb{N}$ uniquely in terms of products in powers of primes:
$$N = 2^{n_2} 3^{n_3} 5^{n_5} \ldots $$ for all $N \in \mathbb{N}$ with $n_i \in \mathbb{N}$
This can be generalized to rational numbers:
$$N = 2^{n_2} 3^{n_3} 5^{n_5} \ldots $$ for all $N \in \mathbb{Q}$ with $n_i \in \mathbb{Z}$
Can this scheme be generalized on more ways?
For example, is not clear that algebraic irrationals are contained in the set of
$$N = 2^{n_2} 3^{n_3} 5^{n_5} \ldots $$ with $n_i \in \mathbb{Q}$
but clearly this set is contained in the algebraic irrationals union rationals.
In a parallel fashion, if we take imaginary unit $i$ as a new element of the prime set, we can rotate in any dense direction in the complex plane by taking all rational exponents
One in principle could define a family of sets in a recursive way:
$$\mathbb{T}_{i+1} = \big\{ N = 2^{n_2} 3^{n_3} 5^{n_5} \ldots n_j \in \mathbb{T}_i \big \} $$
with $\mathbb{T}_0 = \mathbb{Z}$
And ask, do transcendental numbers like $\pi$ or $e$ belong to $\mathbb{T}_i$ for some $i$?