I can think of three. Saul Kripke quite famously could only be begrudged to finish his undergraduate degree at Harvard before being hired as a full professor. Donald Martin (the set theorist of Martin's Axiom fame) seems to have been hired into professorship out of a post-baccalaureate fellowship, without pursuing graduate study. Finally, I can't find evidence that Chaitin ever got any graduate degrees.
Are there other famous examples of great logicians or mathematicians making it into the academy while skipping over the PhD? Examples from before the era of the modern academic degree structure are not interesting for the purpose of this inquiry.
One of my professors told me that these days, someone in a similar situation would likely be put into a no-effort PhD program until the residency requirements are met, with some important publication or another of theirs being declared a "dissertation." None of the professors I asked this to could think of a fourth example. Thus, it seems the no-doctorate logician is a dying breed, one worth examining and cataloguing.