Your $H$-machine concept is essentially the same as the concept of oracle computation, due originally to Turing, which gave rise to the elaborate theory of Turing degrees in computability theory. The idea in that subject is that a machine with oracle $B$ is allowed to make membership queries of $B$ during the course of computation, and an oracle $A$ is said to be computable relative to $B$, written $A\leq_T B$, if membership in $A$ is computable from a machine with oracle $B$.

The philosophy of the subject is that the concept of relative computability provides a measure of the relative information content of two oracles. In this way, the structure of the Turing degrees themselves can be viewed as the structure of the possible information contents of infinite sets. If $A\leq_T B$, then $B$ has at least as much information content as $A$, since any query to $A$ can be replicated via a computation involving queries to $B$. In particular, if $B$ itself is computable, then all such queries to $B$ can ultimately be simulated by a computational procedure, and so $A$ also will be computable. It follows, therefore, that minimal (trivial) Turing degree is the equivalence class consisting of all decidable sets.

In your case, the reals you mention $e$, $\pi$, $\sqrt{2}$ are decidable reals, and even polynomial time decidable, and so having them as an oracle provides at most a polynomial time speed-up in computational power. In the sense of computability theory, as opposed to complexity theory, this is a neglible improvement.