We ain't forgetting the roots!

$\begin{aligned} A & = \int_0^{22} \sqrt{x\sqrt[3]{x\sqrt[4]{x\sqrt[5]{\cdots}}}} dx \\ & = \int_0^{22} \left(x\left(x\left(x\left(\cdots \right)^\frac 15\right)^\frac 14\right)^\frac 13\right)^\frac 12 dx \\ & = \int_0^{22} x^{\frac 12 + \frac 1{3!}+\frac 1{4!}+\frac 1{5!}+\cdots} dx \\ & = \int_0^{22} x^{e-2} dx \\ & = \frac {x^{e-1}}{e-1} \bigg|_0^{22} = \frac {22^{e-1}}{e-1} \approx \boxed{118} \end{aligned}$