As in the Green Acres/Petticoat Junction one; or the Adam 12/Emergency/Dragnet one … so on. Obviously spin offs count.
Which universe has had the most member shows?
They just did a SUPERGIRL/FLASH/ARROW/LEGENDS OF TOMORROW crossover.
Happy Days begat Laverne & Shirley, Blansky’s Beauties, Mork & Mindy, Out of the Blue, Joanie Loves Chachi, The Fonz and the Happy Days Gang, and Laverne & Shirley with Special Guest Star the Fonz . The last two were cartoons.
Happy Days itself started as a single feature on the comedy anthology Love American Style, so you could count all eight shows from that.
All In The Family had five spinoffs and two spinoffs-from-spinoffs, although the show 704 Hauser was actually a spinoff of the Bunker’s house.
All in the Family/Maude/Good Times/The Jeffersons/Checking In/Gloria/Archie Bunker’s Place/704 Hauser.
There’s 8, though 3 of those only lasted a handful of episodes.
First in the thread to mention Tommy Westphall.
First in the thread to provide a link.
shakes tiny fist in impotent rage…
The Westphall universe linked St. Elsewhere, Homicide: Life on the Street, the X-Files, and the entire Law & Order franchise.
Star Trek has 7 TV series (TOS, TAS, TNG, DS9, Voyager, Enterprise, Discovery) all set in the prime universe. (Plus the alternate universe movies, the prime universe movies, and various spinoff games, comics, and books in both universes.)
Okay “The Tommy Westphall Universe” can be the ring to rule them all so let’s try define the question narrowly enough to avoid that mess.
Let’s say in a way that could be reasonably considered (to sitcom level standard) internally consistent?
This thread can’t go on without discussion of Detective John Munch, as portrayed by Richard Belzer.
Originating on Homicide: Life on the Street, he also appeared in several episodes of Law & Order, plus a number of its spinoffs. Add to that a couple of Law & Order spinoffs in which the character has not made an appearance, but which are definitely in the same universe, and several other shows which did crossovers with or featured characters from *those *shows, and we have:
Homicide: Life on the Street
The Wire
St. Elsewhere
Law & Order
Law & Order: SVU
Law & Order: Criminal Justice
Law & Order: Trial by Jury
Law & Order: LA
New York Undercover
Conviction
Deadline
In Plain Sight
Chicago P.D.
Chicago Fire
…and then add in the list of apparently unrelated shows in which Detective Munch has appeared, or been referenced by name:
The X-Files
The Beat
Arrested Development
30 Rock
Unbreakable Kimmy Schmidt
Luther
You could make a case for the Bob Newhart show since the psych ward held a patient called Elliott who said some quack from Chicago put him there. And Cheers since there was a crossover at the end of one episode.
The same universe which contains separate universes. 
The crossover universe sometimes referred to as the Tommy Westphall Universe (or the John Munch Universe) is the group on this page that’s called Group 2. It has (I think) 95 shows in it. Whenever a character appears on two shows played by the same actor, the shows are considered to be part of the same crossover universe:
IIRC, we only ever see an episode of LAW & ORDER getting filmed, or playing on a television set, or whatever, on 30 ROCK; it’s not like Munch shows up and flashes a badge and uses his police training to interrogate the regulars as a character in the same universe; it’s just Richard Belzer, an actor who sometimes plays that part.
Also, it slipped my mind that CONSTANTINE is set there, too.
A Muppet version of John Munch also appeared in a Sesame Street episode.
His character also appears in Spider-Man/Deadpool #6
Star Trek is now linked to the Marvel Universe: Star Trek IV Now Exists in the Same Universe As All Marvel Films Thanks to a Special Cameo - Reactor
It’s already been done.
Three X-Men/Star Trek crossovers in the 1990s.
Also, Star Trek is connected to Larry Niven’s Known Space books since the Kzinti show up on the animated series.
And Discord is obviously Q.
And Niven himself wrote the episode, The Slaver Weapon.
The Vixen cartoon series, if we stretch it slightly.
If we count Netflix then the MCU will, by the end of the year, have six Netflix series (Jessica Jones, Daredevil, Iron Fist, Luke Cage, Punisher, The Defenders), three TV series (the cancelled Agent Carter, Agents of SHIELD, Inhumans) and three more in development (Cloak & Dagger, The New Warriors and whatever they’re doing with Hulu.
That’s twelve. Not bad.
Don’t forget that The X-Files has two other series in the same Universe, the short-lived spin-off The Lone Gunmen, and there was also a one episode crossover with Chris Carter’s other series, Millenium.