One common slogan is that ultrapower arguments rely on the axiom of choice. This isn't as true as it may sound, though. The technique still has purchase in other contexts, such as inner models where determinacy holds. For example, assuming large cardinals every set of reals in $L(\mathbb{R})$ is "tame" in various nice senses; in particular, the Martin measure on the Turing degrees is actually a countably complete ultrafilter. Kanamori discusses applications of this measure and its relatives; for a recent example, Patrick Lutz and Benjamin Siskind used it to make progress on Martin's Conjecture about functions on the Turing degrees.

(Since Patrick is a frequent contributor here, I hope he'll expand on this!)

One common slogan is that ultrapower arguments rely on the axiom of choice. This isn't as true as it may sound, though. The technique still has purchase in other contexts, such as inner models where determinacy holds. For example, assuming large cardinals every set of reals in $L(\mathbb{R})$ is "tame" in various nice senses; in particular, the Martin measure on the Turing degrees is actually a countably complete ultrafilter. In The higher infinite, Kanamori discusses applications of this measure and its relatives; for a recent example, in Part 1 of Martin's Conjecture for order-preserving and measure-preserving functions, Patrick Lutz and Benjamin Siskind used it to make progress on Martin's Conjecture about functions on the Turing degrees.

(Since Patrick is a frequent contributor here, I hope he'll expand on this!)