Let x denote the value of the nested radical in the title: x = sqrt(4 + ³√(4 + ⁴√(4 + …))).

If we define the tail recursively by a_k = k-th root of (4 + a_{k+1}), for k = 2,3,… then x = a_2 and the recurrence is a_k^k = 4 + a_{k+1}. Unlike the multiplicative case you wrote (which is linear after taking logs and gives 4^{e−2}), this additive nesting is genuinely nonlinear: ln a_k = (1/k) ln(4 + e^{ln a_{k+1}}), so there is no known simplification to an elementary constant or a simple expression in terms of e.

Numerically, one can evaluate x by backward iteration: fix a large N, set a_{N+1} ≈ 1 (or more accurately a_{N+1} ≈ exp[(ln 5)/(N+1)]), and compute a_N = (4 + a_{N+1})^{1/N}, a_{N-1} = (4 + a_N)^{1/(N-1)}, …, a_2. This converges rapidly and is insensitive to the seed. Doing so gives x = a_2 ≈ 2.4015 (more precisely, about 2.4015 to 4 decimal places).

A simple analytic lower bound comes from convexity of y ↦ ln(4 + e^y): ln a_k ≥ (1/k)(ln 5 + (1/5) ln a_{k+1}), which, unrolled, yields ln x ≥ ln 5 · ∑_{m=0}^∞ 1/(5^m (m+2)!) = ln 5 · (e^{1/5} − 1 − 1/5)/(1/5)^2, giving x ≥ exp(≈0.861…) ≈ 2.366. Tighter bounds are obtainable numerically as above.

Conclusion: unlike the product-nesting which equals 4^{e−2}, the sum-nesting sqrt(4 + ³√(4 + ⁴√(4 + …))) does not have a known “nice” closed form; its value is approximately 2.4015.