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.