In my answer to a question about degrees of irrationality, I had posted the following summary account of degrees of complexity, which in the end I believe ultimately reaches into the realm of the upper hierarchies of vastness to which you seem to aspire.

The other answers and comments are fascinating, particularly about the irrationality measure, but allow me to give a little more information along the lines of Mark Sapir's answer by mentioning that there are several very large, intensely studied hierarchies of complexity for reals numbers. After the initial familiar notions come several others...

• rational

• algebraic

• computable

The computable reals are those for which we can compute rational approximations to any desired accuracy, by Turing machine. (A concept used in computable analysis.) The computable subsets of $\mathbb{N}$ are those for which we can compute yes/no answers for membership in finite time. For example, all the numbers you mention in the question, such as $\pi$ and $e$, are computable.

• computably enumerable

The c.e. subsets of $\mathbb{N}$ are those for which there is a computable enumeration procedure. Equivalently, you can compute the yes answers for membership in finite time. The concept of relative (oracle) computability leads to the hierarchy of Turing degrees, which measures the comparative computable complexity of a real.

• arithmetic

A real $x$ is arithmetic if it's digits can be defined by a definition involving only quantification over the natural numbers and primitive operations. Equivalently, the arithmetic subsets of $\mathbb{N}$ arise from the computable subsets of $\mathbb{N}^k$ by projection and complement. The arithmetic hierarchy breaks naturally into levels, such as $\Sigma^0_n$ and $\Pi^0_n$, corresponding to the logical complexity of these definitions, and these levels are refined by the Turing degrees. For example, the set of Turing machine programs $p$ which compute total functions forms a complete $\Pi^0_2$ set. The relativized notion leads to the arithmetic degrees.

• hyperarithmetic

A real is hyperarithmetic if it can be defined by two equivalent definitions, one involving just one universal quantifier over the reals and another having just one existential quantifier over the reals, and otherwise any level of arithmetic quantifiers. This is the same as $\Delta^1_1$. The hyperarithmetic hierarchy is stratified in a hierarchy of length $\omega_1^{CK}$, a lightface version of the Borel hierarchy, in which one uses uniformly computable countable unions and complements. The relativized notion leads to the hyperarithmetic degrees, a hyperarithmetic analogue of the Turing degrees.

• projective

A real is projective if it can be defined by a description that quantifies only over the set of real numbers, plus natural number quantification and the primitive operations. The projective hierarchy is stratified by considering the logical complexity of these definitions, with levels $\Sigma^1_n$ and $\Pi^1_n$. For example, the lightface analytic sets are $\Sigma^1_1$ and co-analytic is $\Pi^1_1$, with hyperarithmetic being $\Delta^1_1=\Sigma^1_1\cap\Pi^1_1$.

• constructible

A real is constructible if it exists in Gödel's constructible universe $L$. The concept of relative constructibility gives rise to the constructibility degrees, by which $x\sim y\leftrightarrow L[x]=L[y]$, forming a rich hierarchy.

• ordinal-definable

A real (or set) is ordinal-definable if there is a definition of it in the language of set theory, using ordinal parameters. For example, the real whose $n^{th}$ binary digit is $1$ just in case $2^{\aleph_n}=\aleph_{n+1}$ is ordinal definable. The class HOD of all hereditarily ordinal definable sets satisfies ZFC, but can be strictly smaller than the universe of all sets.

• generic

A real is generic over $L$ (or some other fixed universe $V$) if it exists in a forcing extension of $L$ (or $V$) by set forcing. Of course, it is relatively consistent with ZFC that every real is generic over $L$, since this is true in $L$ itself, but under some large cardinal axioms, there are reals, such as $0^\sharp$, that cannot be added by forcing over $L$.

The higher levels of these latter hierarchies are further developed and stratified by the enormous variety of models of set theory arising from large cardinals, various inner model constructions, forcing extensions and so on, so that the hierarchy loses its linear nature, becoming instead a dense jungle of various interacting concepts of set theory.