Bad experiences with the production of an accepted manuscript The long and erroneous publishing process after a recently accepted  (Open Access) paper has led me to ask this question. I'll explain its context briefly.
We submitted a journal paper (a LaTeX manuscript including supplemental material) which after, in my domain (Computer Science) somewhat reasonable, 7 to 8 months time got accepted subject to minor revision. The journal has a very good reputation, the reviewers unanimously liked the manuscript and gave very helpful and detailed feedback. I handed in the revision not quite a month later. About 4 months from then I got the final acceptance notification. Although I felt already that took quite long, what followed seems like an Odyssey to me: after more than two months I got the first online proof which had numerous (100s of) typos, type setting and conversion errors introduced, tables wrongly formatted, cross-references lost, etc. I had to invest considerable time to point out all the errors, and after several more weeks (about two months in the meanwhile) and two proof revisions, I'm still stuck with waiting for another iteration of the proof. The editorial office has politely and helpfully handled the corrections with the production team, however, my requests/concerns also got a bit lost in translation. Overall, we are about to reach 9 months after acceptance with minor corrections and, after a couple of emails with the editorial office, I have no idea how long it will still take to at least see something like an official online-first version, not talking of the journal issue the article will possibly be included.
I believe that journal's procedure seems inappropriate inasmuch as I opted for full Open Access according to the funding policies and my Uni accepted (by a default waiver for that publisher) to pay a decent amount of article processing charges (I believe from their library budget). This inappropriateness stems from the long production time and that the conversion of the LaTeX manuscript caused so many further issues and required several proof iterations. Maybe strange, but again, I would have expected a more accurate production process than for a non-OA publication. But OA might not play a role here.
Just for comparison: a year earlier, we had a non-OA article accepted with minor for another (even higher ranked) journal where it took them about six weeks from this acceptance to complete publication, no typos introduced, and those six weeks even included our time to submit the minor revisions.
That whole thing made me wonder what expectations one ought to have on the paid portion of work done in the production process, other than reviewing and editing which in the majority of cases is purely voluntary in my field? How could one improve this situation? E.g. would a notification of my Uni (maybe our library) be exaggerated, so they can consider blocking future APC waivers for that journal/publisher?
I believe my concerns go beyond this interesting question about long review processes.
It doesn't matter whether you paid an APC or whether the publisher is getting money from subscribers for seeing your article.
There are journals that have very significant production backlogs, so that accepted papers are not produced for months or even years.  A long production backlog is not really cause for serious complaint, though undoubtedly both you and the editors of the journal are not happy about it.  Note that frequently the only way to fix a production backlog is to stop accepting papers (or accept very few papers) for some period of time.  (In particular, back when people mostly read papers by going to their physical library, there was no reason for the production backlog to be any better than the publication backlog, and some publishers still think this way.)
However, the number of errors in the production process and the amount of time and number of iterations it has so far required to fix them is completely unheard of in my experience.  I have had some production issues twice in 20+ papers (both times with the same publisher but different journals), but they were nowhere near what you are describing.
At this point, you should let the editors know what has happened and ask them to complain strongly on your behalf.  (Of course, you should be nice and understanding about it, since it's not their fault in any way.  But they have leverage that you don't.)  They are on your side.  They can see you have been treated badly, and they know that incidents like this are very bad for their journal.  This should be a serious issue, to the extent that, if these kinds of production issues are a common occurrence, the editorial board might try to switch publishers (if they own the name of the journal) or resign.  (You don't know if your case is a one-off or common, so you can't very well ask them to do this, but they should want to know and should take this seriously.)
Finally, I think you are well within your rights to not submit to a journal run by this publisher until you have some indication they have improved their processes, and you should quietly let anyone who asks you for journal advice know about what you have gone through.  (If someone reads this question, figures out who you are, and e-mails you, you should let them know who this publisher is.)