So why bring some numbers here as transcendental if not proven?

For context, a number is transcendental if it's not the root of any non-zero polynomial with rational coefficients. Essentially, it means the number cannot be constructed using a finite combination of integers and standard algebraic operations (addition, subtraction, multiplication, division, and integer roots). sqrt(2) is irrational but algebraic (it solves x^2 - 2 = 0); pi is transcendental.

The reason we haven't been able to prove this for constants like Euler-Mascheroni (gamma) is that we currently lack the tools to even prove they are irrational. With numbers like e or pi, we found infinite series or continued fraction representations that allowed us to prove they cannot be expressed as a ratio of two integers.

With gamma, we have no such "hook." It appears in many places (harmonics, gamma function derivatives), but we haven't found a relationship that forces a contradiction if we assume it is algebraic. For all we know right now, gamma could technically be a rational fraction with a denominator larger than the number of atoms in the universe, though most mathematicians would bet the house against it.

The human-invented ones seem to be just a grasp of dozens man can come up with.

i to the power of i is one I never heard of but is fascinating though!

1: Almost all numbers are transcendental.

2: If you could pick a real number at random, the probability of it being transcendental is 1.

Most of our lives we deal with non-transcendental numbers, even though those are infinitely rare.

The transcendental number whose value matters (being the second most important transcendental number after 2*pi) is ln 2 = 0.693 ... (and the value of its inverse log2(e), in order to avoid divisions).