  Let B_1,B_2, ... be independent one-dimensional Brownian motions defined over
the whole real line such that B_i(0)=0. We consider the nth iterated Brownian
motion W_n(t)= B_n(B_{n-1}(...(B_2(B_1(t)))...)). Although the sequences of
processes (W_n) do not converge in a functional sense, we prove that the
finite-dimensional marginals converge. As a consequence, we deduce that the
random occupation measures of W_n converge towards a random probability measure
\mu_\infty. We then prove that \mu_\infty almost surely has a continuous
density which must be thought of as the local time process of the infinite
iteration of independent Brownian motions.
