The following is known (see e.g., [1]): Modulo large cardinals it is consistent that $2^\lambda>\lambda^+$ for all $\lambda$. (Furthermore, it seems that the gap can be always infinite?)

Question: Is it consistent that $2^\lambda=2^{\lambda^+}$ for all $\lambda$? (Which obviously implies infinite gap.)

Remark: For just the regulars this can be done for free by Easton, e.g. setting $2^{\aleph_\alpha}=\aleph_{\alpha+\omega+1}$.

[1] Foreman, Matthew; Woodin, W. Hugh, The generalized continuum hypothesis can fail everywhere, Ann. Math. (2) 133, No. 1, 1-35 (1991). ZBL0718.03040.