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Let $W$ be a two sided real valued Brownian motion. Let $B$ be a one sided Brownian motion independent of $W$. Consider the process $X(t)=W(B(t))$. Is the quadratic variation finite and if it is, what is the quadratic variation of $X$ on $[0,T]$?

I am just curious if we can iterate processes like this.

More generally, given $n$ independent two sided Brownian motions $B_1,...,B_n$ does the quadratic variation of the process $X^n(t)=B_1(B_2(...B_n(t)))$ exist on $[0,T]$?

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turning comment into answer.

This process appears here "Stochastic Calculus for Brownian Motion on a Brownian Fracture" and they do some quadratic calculations there.

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