Has anyone else thought about this? I've browsed the Wikipedia for mathematical constants and only things I saw that were definitely rational AFAIK were 0, 1, 2, and 1/2, which don't really count.
I think 777480/8288641 counts also. Fermat's/ Euler's Diophantine problem.
Are there any questions which have been asked where a rational answer is expected to be the answer?
Seeking tags also because I'm unsure what area of math this relates to most.
From Wikipedia, Legendre's constant was first conjectured to be $1.08366...$ but later proved to be exactly $1$.
There are whole sequences of important rational numbers. See these pages on "integer" sequences (although they may have some non-integer rational values, e.g. the earliest Bernoulli numbers, which are the basis for results such as the famous $-\tfrac{1}{12}$ result @jojobo mentioned) & this page for rational ones.
Then there are quantities of the form $\zeta(2n)/\pi^{2n}$ with $n$ a positive integer.
One could argue every rational $s$ is important for a suitably chosen complex analysis problem due to factors of $e^{2\pi is}$ or $(1-e^{2\pi is})^{\pm1}$.
