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] <cite authors="Foreman, Matthew; Woodin, W. Hugh">_Foreman, Matthew; Woodin, W. Hugh_, [**The generalized continuum hypothesis can fail everywhere**](https://doi.org/10.2307/2944324), Ann. Math. (2) 133, No. 1, 1-35 (1991). [ZBL0718.03040](https://zbmath.org/?q=an:0718.03040).</cite>