A recursively-defined process can also be a martingale, but adding a deterministic drift breaks it

Let $Y_1,Y_2,\dots$ be a sequence of independent random variables with mean zero and let $\mathbf F_n=\sigma(Y_1,\dots,Y_n)$. Define $X_n=\sum_{k=1}^nY_k$; $W_1=Y_1$ and $W_{n+1}=W_n+Y_{n+1}W_n^2$; and $T_n=\sum_{k=1}^n(Y_k+1)$. Then: