Hoping it is not too famous an open problem, I would suggest trying to (dis)prove that Euler's constant $\gamma$, defined as $\displaystyle{\lim_{n\to\infty}H_{n}-\log n}$ where $H_{n}$ is the $n$-th harmonic number, is irrational. A plausibly interesting approach may rely on Hankel determinants, that were successfully used by Yann Bugeaud et al to obtain new results about irrationality exponents in http://arxiv.org/abs/1503.02797.