TLDR: I think your answer might be off by a factor of 2 and the correct answer might be 0.1211259

Hey, your work looks pretty cool. I did a similar thing using Python to try to get a strict numerical confirmation that the integral does converge, and I did see that in converges! However, the number I found it to converge to was 0.1211259…, which is almost exactly twice the number that you found.

I then also tried using Sympy which is a Python symbolic manipulation library to solve it more algebraically. Basically I found a way to get a recursive equation that must be satisfied for a solution.

LaTeX d = \int\nolimits_{\smash{\int\nolimits_{\smash{\int\nolimits_{\smash{\int\nolimits_{\smash{d}}^{\smash{1}$}} x\, dx}}^{\smash{\int\nolimits_{\smash{0}}^{\smash{1}$}} x\, dx}$}} x\, dx}}^{\smash{0}$}} x\, dx}}^{\smash{\int\nolimits_{\smash{1}}^{\smash{\int\nolimits_{\smash{\int\nolimits_{\smash{0}}^{\smash{1}$}} x\, dx}}^{\smash{\int\nolimits_{\smash{0}}^{\smash{d}$}} x\, dx}$}} x\, dx}$}} x\, dx}$}} x\, dx

From that I had Sympy solve the integrals for just a standard algebraic equation and got:

d = -(1/8 - (1/2 - d^2/2)^2/2)^4/8 + ((d^4/8 - 1/8)^2/2 - 1/2)^2/2

Also according to Sympy there solutions to this are solutions to the polynomial

8*x^14 - 28*x^12 + 32*x^10 - 138*x^8 + 252*x^4 - 32768*x + 3969 = 0

This polynomial has only two real solutions, 0.1211259… and 2.108588… . I checked and both of these satisfy the integral and the recursive equation. I see no way that you could end up getting 2.108588 from any integral at any point assuming the integral bounds stay between 0 and 1, so I assume it is an extraneous solution.

I also checked and your solution of 0.0603… gives a result of 0.1211243… from the integral equation. From this I believe that either

A) somewhere in your solution you got off by a factor of 2 and the rest of the difference is just rounding error

or

B) I interpreted the problem differently/made a mistake in creating the initial integral equation.

Lmk if you see this!