Ito's formula for two to the power W

For standard Brownian motion $W$, apply Ito's formula to $Y_t=2^{W_t}$. Is $Y$ a martingale?

No. Writing $Y=e^{W\ln2}$, Ito gives $dY=\ln2\,Y\,dW+\tfrac12(\ln2)^2Y\,dt$. The positive drift means $\mathbb{E}[Y_t]=e^{(\ln2)^2t/2}$ grows, so it is a submartingale; the rescaled $2^{W_t}e^{-(\ln 2)^2t/2}$ is a martingale.