In a polynomial of $n$ degree, what numbers can fill the $n$? Until now, I've seen that the $n$ could be filled with the set $\mathbb{N}_0$ and $-\infty$ but I still didn't see mentions on other sets of numbers. As I thought that  having 0 and $-\infty$ as degrees of a polynomial were unusual, I started to think if it would be possible for other numbers to also fill the gap.

The degree of a polynomial can only take the values that you've specified. For that, let's revisit the definition of a polynomial. Personally, I was taught that a polynomial (in one variable) is an algebraic expression which can be written in the form $$a_0 + a_1x + a_2x^2 + \cdots + a_nx^n$$ where $n$ is a non-negative integer. From this definition, as the degree is the same as $n$, therefore, the degree is also a non-negative integer (I'll not go into the degree of a zero polynomial, which is best left undefined).
On the other hand, Wikipedia says that 

A polynomial is an expression of finite length constructed from
  variables (also called indeterminates) and constants, using only the
  operations of addition, subtraction, multiplication, and non-negative
  integer exponents.

Clearly, there is no non-integer expenentiation, nor any division by a variable, so again it is clear that the degree is a non-negative integer.
