I can usually at least finish any beer, even the worst of the worst. If I've opened it, I'll finish it although wouldn't necessarily have another one if the beer is that bad. But there's two occasions in my life I've really struggled.
First occasion when I was a student on very little money. A mate came round for a session and we bought as much of the cheapest beer we could afford. Co-op Bavarian Lager (Co-op being the Cooperative stores in the UK). It worked out at around 50p a can (this was around 20something years ago). "How bad can it be?" It was terrible. So terrible in fact, and we were determined to finish off the 16 cans we'd bought, that we scraped together what little money we had and ran round the corner to the off license to pick up a cheap bottle of gin or whisky to down with the beer to take away the taste.
Second beer that I had trouble finishing off was at a Chilli Festival in Brighton, UK last year. It was a chilli beer and absolute muck. Three or four swigs and I was done. I think that was probably the only beer I've not actually been able to finish off. I've had a couple of chilli beers in the past and not one of them was of any reasonable quality and something I'd ever buy again. I just don't think chilli works in beer although in theory it should work great.