Does $A_n= \sqrt{1^2+\sqrt{2^2+\sqrt{...+\sqrt{n^2}}}}$ converge?

See here for references to a criterion. $$\limsup_{n\to\infty} \frac{\log n^2}{2^n} = 2\limsup_{n\to\infty} \frac{\log n}{2^n} = 0$$ So the convergence is positive. The value is another story though.