The DU Lounge

Let's look at Travis McGee. 21 books, as you say, all of which I've read multiple times and several of which I love. But is it really a "series," or just a barely-connected sequence of mostly-unrelated incidents? I tried to compile a timeline for Travis's career and ended up with the sad conclusion that there is virtually no real connection among the books. The exception being the last, written when he was checking out and he wanted to tie up the series for his fans (laudable decision by a dying man), and ended up a very touching last episode.

One of the problem long-running series have, especially mystery/thriller ones, is that they are written over a space of decades, and therefore the character who was young and capable in the first books should be an old, tottering incompetent by the time the last book is written. How old was Travis when he met Lois? How old when Jean shows up? John did occasionally reference these problems -- Travis makes occasional noises about "Birthdays with a Zero" and slowing down/needed more time to recover from adventures, but there's no way he ages 20 years between the first book and the last. Other authors have adopted different devices; one I know sets every book according to a strict timeline, which means books written in 2020 are set in 1990, making them a very odd sort of historical fiction.

That said, to address your question:

In addition to Travis McGee, I've become "involved" with so many series that they'd be impossible to list. Favorites include:

Flashman series by George MacDonald Fraser. Historical comedy with footnotes. Fraser was the most popular author in Britain for awhile before Terry Pratchett overtook him.
Discworld series by that selfsame Terry Pratchett. Not really a connected "series" at all, but several sequences involving particular casts of characters who all exist in a (very) loose timeframe on the Discworld.
Black Company series by Glen Cook. The long story of a company of quasi-mediaeval/fantasy mercenaries who start out in the service of the bad guys, and go downhill from there. Revolutionary for the genre for many reasons. Mr Cook is considered the father of the gritty, realistic sort of fantasy that GRR Martin has made bank on.
Dread Empire series by ditto. This is fantasy on a higher plane than the Black Company, at the level of the clash of empires rather than the day-to-day struggles of a group of grunts trying to stay alive. Unlike, say, the Silmarillion, though, it does have viewpoints from the individual level which drive the narrative, so it is not a "fantasy history" like that book.
Garrett, P.I. series by ditto. And now for something completely different: a genre-blending series that combines a fantasy setting with a noir P.I. protagonist. A tribute to John D. MacDonald in that all the books have titles with metals in them: eg, Sweet Silver Blues, Cold Copper Tears. Also intended to be humorous, if the combination of elves and Sam Spade didn't make that obvious.
The Chronicles of Amber, by Roger Zelazny. In two five-book arcs, a fantasy series that also adopts a kind of gritty, noir approach, and also has echoes of John D. MacDonald (whom Roger directly references in one book). Another parallel with MacDonald is that Roger wrote the last volume of the series literally on his deathbed, in an attempt to tie up all the loose ends for his fans. In justice, he fails and it is unsatisfactory, but as he was busy dying at the time, one can hardly hold it against him.
The Vorkosigan Saga, by Lois McMaster Bujold. Space-opera sci-fi with a unique protagonist. Bujold won the Hugo four times for novels in the series, and I personally believe it was only four times because the Hugo committee didn't want to break Robert Heinlein's record. Lots of comic elements in the series, and my favorite novel of the whole 17 or so is A Civil Campaign which is outright subtitled "A comedy of manners."
The Baroque Cycle + Cryptonomicon, by Neal Stephenson. Not strictly a series, a long trilogy with a fourth volume that is closely related to the trilogy. Stephenson combines history, philosophy, fantasy, and information science in a strange brew which is definitely not everybody's cup of tea, but since I am on a first-name basis with many of the historical characters he references (Leibniz, Newton, Hooke, eg), I love it to pieces.

There are many more examples that could be cited, but this post is already threatening to become longer than the subject matter, so I'll stop here.

-- Mal