Here I mean the limit of the following sequence:
$$p_1=\int_0^1 \sqrt{x} ~dx=\frac{2}{3}$$
$$p_2=\int_0^1 \int_0^1 \sqrt{x+\sqrt{y}} ~dxdy=\frac{8}{35}(4 \sqrt{2}-1) = 1.06442\dots$$
$$p_3=\int_0^1 \int_0^1 \int_0^1 \sqrt{x+\sqrt{y+\sqrt{z}}} ~dxdydz = 1.242896586866\dots$$
$$p_4 \approx 1.314437693607766$$
$$p_5 \approx 1.34186271753784$$
Here the approximate values are computed by Mathematica. In principle every one of these integrals can be evaluated in closed form, but it becomes very complicated (see $p_3$ at the bottom of the post).
How can we find the limit at $n \to \infty$? It should be finite because of the range of variables chosen.
$$\lim_{n \to \infty}p_n=\lim_{n \to \infty} \int_0^1 \cdots \int_0^1 \sqrt{x_1+\sqrt{x_2+\sqrt{\dots+\sqrt{x_n}}}}dx_1 dx_2\dots dx_n=?$$
I find it very likely that $\lim_{n \to \infty}p_n=\phi$ (the Golden Ratio), but I'm not sure (this is not correct, see the comments).
Edit: With the help of Wolfram Alpha I tackled $p_3$ (see the updated numerical value above):
$$p_3=\frac{64}{135135} (2 \sqrt{3244081+2294881 \sqrt{2}}-664\sqrt{2}-1092\cdot 2^{3/4}+305)$$
This confirms my suspicions that there is no hope for apparent pattern in the first few $p_k$. Now an interesting challenge is to see how many $p_k$ can be realistically computed in closed form.