I would assume, after 25+ years of inter-league play with 30 teams, pretty much every team has beaten every single other MLB team. To take a sample pairing, Cubs vs. Blue Jays, we have 16 games, which even if one team is expected to win every game 2/3 of the time, has a .15% chance of being a sweep.
With 15 teams in each league, and assuming intra-league series are way too long to be a sweep, that gives a (1-.0015224)^(15*15)=70.98% chance everyone has beaten everyone, and I'm making a big assumption about how one-sided series might be.
So I'd be curious about the first and last teams, and everyone in between. This seems possible by data-scraping the site above (or Retrosheet, where they got it from) and parsing each page for a "W" and finding the date near it, which would be a neat Python project, but I can't make the time for it just now.
So does anyone know, or can anybody figure, the dates each MLB team finally beat everyone else? (I may answer this myself if I have the time this weekend. But I wanted to throw the question out there.)