What will happen if Brownian motion is taken iteratively (i.e. W(W(t)), W(W(W(t))), ...)?

Question

We consider a one dimensional bi-directional standard Brownian motion $\{W(t)\}_{t\in\mathbb{R}}$ with $W(0) = 0$. Then what can we say about $\{W(W(t))\}$? Is it a martingale? Is it Markovian? And what about $\{W(W(W(t)))\}$, $\{W(W(W(W(t))))\}$, and so on?

I tried a simple MATLAB algorithm to simulate $W^{(K)}(t)$. It seems that the process converge to something that oscillates and jumps crazily. But I am not sure whether my simulation is correct since I am using a linear interpolation while the Brownian motion is not piecewise differentiable. Therefore, I also want to know: is there any way to simulate these processes more accurately?

clear; clc;
T = 10;
N = 1500;
x = linspace(-T, T, 2 * N + 1);
BM = zeros(1, 2 * N + 1);
for i = 1 : 2 * N
    BM(i + 1) = BM(i) + randn * sqrt(T / N);
end
BM = BM - BM(N + 1);
% plot(x, BM);
BMi = BM;
K = 64;
for j = 2 : K
    BMi_new = interp1(x, BMi, BM);
    disp(norm(BMi_new - BMi))
    BMi = BMi_new;
end
plot(x, BMi);