None? I'm prone to say many.
At least in my younger days I knew of many, including academic journals with which I had close personal involvement, which had very strict requirements. However, I do think it very possible that those rules have been relaxed somewhat in more recent times for some (perhaps even many) academic society journals. For that, one would need to consult someone with more recent experience. So perhaps I should temper my surprise at the reactions to Jon's statement. Even so, I'd wouldn't be surprised if a great many journals still had strict criteria, whether officially stated or not. It may be a de jure versus de facto situation.
As I recall it, there were many journals published by academic societies which may not have had an official published rule "You must hold a PhD in a relevant field and a full-time academic appointment to submit an article for publication in this journal." Yet, in actual fact, one couldn't submit a manuscript unless one had FULL membership in the academic society---and that full membership required an appropriate Ph.D. and nominations from other society members, who almost always required that a permanent academic post be held by the nominee. (I don't recall a teaching post to be necessary, as long as the institution of employment itself was a teaching institution. Thus, a research professor, for example, at such a school was not ineligible just because he/she didn't hold a teaching position.)
At my age, memory doesn't always serve me well. But an example which comes to mind was JBL_, the official journal of the Society of Biblical Literature. I think they have relaxed their membership requirements nowadays in order to bring in more membership revenue. But when I joined long ago there was a very strict set of requirements for full membership. And the requirements for presenting a paper at an academic conference were formidable, including finding a full member (i.e. someone with an appropriate PhD and a full-time academic post at a recognized institution) putting their name on the line as endorsing the paper. Yet, even then, there were a lot of hurdles which one had to overcome including a lot of the "old guard" who could veto the paper. Publishing an article in JBL itself was virtually impossible for someone not meeting the aforementioned requirements.
In fact, I remember a controversy among the JBL powers-that-be as late as the 1970's when an individual who some considered to be analogous to a Jack Horner was getting a significant degree of notice among academics for his work in Assyriology (???). He was not a member of SBL and didn't meet the requirements, but due to someone bending the rules, he had presented a well-received paper at a regional SBL meeting. So there was a debate among full members on whether his paper was qualified for JBL publication, no matter its merits. I can't recall how it panned out.
Admittedly, my memories may come from a very different time. But when I read "That is utterly, completely, spectacularly false, Jon.", I was quite startled---because basically everything @Jonathan_Burke wrote on this topic harmonized with my personal experience as a published academic. Furthermore, their application to a relative "fringe outsider" like Richard Carrier concurs with my memories exactly. Indeed, I had a lot of dealings with both Carrier and Robert Price and their controversial ideas before my retirement. It came to a head when a PBS producer wanted to included them in some local affiliate's plans for a TV documentary on Jesus mythicism. (Carrier and Price were overwhelmingly ignored by the academy at large and were often the target of professional jabs, because neither had managed to find what was considered "credible employment" in the academic world. A PhD alone rarely brings much respect among academics no matter how popular one becomes with the general public.)
Yes, there are prestigious journals for which authors don't need to jump through significant academic achievement hoops. But I think even today there are a great many journals which enforce standards nearly as tight as those of my experience. I say that based on recent anecdotal comments from colleagues even after my retirement.
Of course, my specific example was a humanities related journal and not a scientific one---but I have similar memories with a particular scientific journal involving an academic society in the 1980's. In my experience, there are many similarities in how academics manage their journals, both in the humanities and in scientific fields. (My research was of an interdisciplinary nature and so I had my fair share of experiences on both sides of that divide.)