My original question (posted in https://math.stackexchange.com/questions/1584430/can-all-power-sets-be-limit-cardinals) was:

Is it possible to create a model of ZFC, so that **the cardinality of each power set is a limit cardinal** (as opposed to GCH where they are always successor cardinals)?

Obviously, from Easton's theorem, **this is possible for regular cardinals**.
I also showed that in this model strong limit cardinals must also be fixed points of the $\aleph$ function (using Shelah's upper bound from PCF), and therefore such a model would mean $2^{\aleph_\delta} \geq \aleph_{\delta^+}$. But whether we can improve Shelah's upper limit from $2^{\aleph_\delta} < \aleph_{{|\delta|}^{+4}}$ to $2^{\aleph_\delta} < \aleph_{\delta^+}$ is an open question. So **for strong limit cardinals, we don't know whether such a model is possible** (even for one cardinal, let alone all of them).

**This leaves the singular weak limit** $\delta$ case, which isn't interesting unless we assume the continuum function doesn't become constant for all $\gamma$ such that $\kappa < \gamma < \delta$. **Assuming this, can we create a model of where the cardinality of the power set of singular weak limit are limit cardinals**?

For a concrete example, take the following rule (which satisfies Easton's theorem, and the known limitations for singular cardinals in http://www.math.tau.ac.il/~gitik/icm5.pdf): $$2^{\aleph_\alpha} = \aleph_{\aleph_{\alpha + 1}}$$