$\lim\limits_{n\to\infty}\operatorname{arsinh}(1 + \operatorname{arsinh}(2 + \operatorname{arsinh}(3 + \dots\operatorname{arsinh}(n+\dots)\dots)))=?$

Question

Does the limit

$$\lim\limits_{n\to\infty}\operatorname{arsinh}(1 + \operatorname{arsinh}(2 + \operatorname{arsinh}(3 + \operatorname{arsinh}(4+\dots\operatorname{arsinh}(n+\dots)\dots))))$$

exist and if it exists, is there a closed form for it ?

PARI gives the numerical value :

? n=500;x=n;while(n>0,x=asinh(x)+(n-1);n=n-1);print(x)

1.877794886208648239222988918

?