The simplest generalization of
x = \sqrt{2 + \sqrt{2 + \sqrt{2 + ...}}}
is
x = \sqrt{a + \sqrt{a + \sqrt{a + ...}}},
for some a \gt 0.
Since the sequence
\sqrt{a}, \sqrt{a + \sqrt{a}}, \sqrt{a + \sqrt{a + \sqrt{a}}}, ...
is obviously monotone increasing, we only need to prove its boundedness to be in a position to apply Monotone Convergence Theorem.