Total number of mathematical constants

Question

Can we know how many interesting constants exist in all like $\pi$ and $e$?

Answer 1

There are lots of interesting constants. See Finch's Mathematical Constants, now in its second volume. More here.

Here is the blurb from the first volume:

Mathematical Constants. Famous mathematical constants include the ratio of circular circumference to diameter, π = 3.14 . . . , and the natural logarithmic base, e = 2.178 . . . . Students and professionals usually can name at most a few others, but there are many more buried in the literature and awaiting discovery. How do such constants arise, and why are they important? Here Steven Finch provides 136 essays, each devoted to a mathematical constant or a class of constants, from the well known to the highly exotic. Topics covered include the statistics of continued fractions, chaos in nonlinear systems, prime numbers, sum-free sets, isoperimetric problems, approximation theory, self-avoiding walks and the Ising model (from statistical physics), binary and digital search trees (from theoretical computer science), the Prouhet–Thue–Morse sequence, complex analysis, geometric probability, and the traveling salesman problem. This book will be helpful both to readers seeking information about a specific constant and to readers who desire a panoramic view of all constants coming from a particular field, for example, combinatorial enumeration or geometric optimization. Unsolved problems appear virtually everywhere as well. This is an outstanding scholarly attempt to bring together all significant mathematical constants in one place.

Answer 2

More interesting constant for me is the Euler $\gamma$ constant defined by $$\gamma=\lim_{n\to \infty}\biggl(\sum_{k=1}^n\frac{1}{k}-\ln n\biggr)$$ Between many problems actually unsolved is to Know that this constant is irrational or not, therefore its important to know more about this costant.