According to The Tommy Westphall Universe Hypothesis, nearly all of American television takes place in the mind of an autistic child. The theory first appeared in 2002, when writer Dwayne McDuffie wrote Six Degrees of St. Elsewhere for the Slush Factory.

For those of you don’t know, St. Elsewhere was a slick, well written and acted drama series about the doctors, administrators and patients of the fictional Boston hospital, St. Eligius (nicknamed St. Elsewhere by the staff). After a long, award-winning run, in the very last moments of the show's final episode, it was revealed that all of the events of the show were merely the prolonged daydream of an autistic child. None of it "really" happened. Whether you like this final twist (for what it’s worth, I didn’t), it’s a legitimate ending to a self-contained show. But if St. Elsewhere played by the rules of comics, either they wouldn’t have been allowed to do it, or they would have precipitated a crisis in TV Land far bloodier than DC Comics' Crisis On Infinite Earths. Why? Because crossover-wise, St. Elsewhere is the Kevin Bacon of TV shows.

Stay with me now, this is complicated but kind of fun.

Characters from St. Elsewhere have appeared on Homicide, which means that show is part of the autistic child’s daydream and likewise doesn’t exist. It gets worse. The omnipresent Detective John Munch from Homicide has appeared on X-Files, Law & Order and Law & Order: SVU. Law & Order characters have appeared on Law & Order: Criminal Intent. X-Files characters have appeared on The Lone Gunmen and Millennium. Characters from Chicago Hope have appeared on Homicide. Characters from Picket Fences have appeared on Chicago Hope. All those shows are gone (if you count cartoons, which makes this game much too easy, the X-Files characters have appeared on The Simpsons. The Critic has also appeared on The Simpsons. Dead).

Characters from Picket Fences have appeared on Ally McBeal. Ally McBeal has appeared on The Practice. Characters from The Practice have appeared on Boston Public. Autistic daydreams, every one.

But that's not all. St Elsewhere characters have appeared on Cheers, so Fraiser doesn’t exist. Neither do Wings, Caroline In The City or The Tortellis but who cares? Well, maybe you do, because Caroline In The City once crossed over with Friends, which crossed over with Mad About You, which crossed over with Seinfeld and The Dick Van Dyke show. None of them happened in our new, shared continuity.

St. Elsewhere also shared characters with The White Shadow and It’s Gary Shandling’s Show. Garry Shandling crossed over with The Andy Griffith show (no, really!). So Gomer Pyle, Mayberry RFD, and Make Room for Daddy/The Danny Thomas Show are gone. Make Room For Daddy takes out I love Lucy.

And there’s more, St. Elsewhere also shares continuity with M*A*S*H, so Aftermash and Trapper John MD are out of there.

Now here’s a good one, St. Elsewhere shared a patient with The Bob Newhart Show, so the Bob Newhart Show is part of the grand daydream. The Bob Newhart Show crossed over with Murphy Brown, which in turn links to, among many others: Julia, The Nanny, Everybody Loves Raymond and I Dream of Jeannie! Meanwhile, the series Newhart was revealed to be a nightmare had by Bob Newhart’s character on the Bob Newhart Show. Newhart crossed over with Coach, which connects it to Grace Under Fire, Ellen, and Drew Carey. Drew Carey takes out Home Improvement and NYPD Blue.

All of these shows (and many more that I left out or missed) are daydreams of St. Elsewhere’s autistic kid.

That's forty-seven shows, by my count, that exist in this shared universe. You can also add Boston Legal, which shares many characters with The Practice, and both The Single Guy and Hope and Gloria for their crossovers with Friends to bring the count to an even fifty. Wait, no. Joey makes fifty-one. (We can keep adding to the list, since many shows that came out after 2002 cross over with shows already mentioned.)

Detective John Munch, played by the same actor, Richard Belzer, has appeared in at least a single episode of the following distinct television series, though in some cases, multiple episodes:

"Law & Order: Special Victims Unit"
"30 Rock"
"The Wire"
"Arrested Development"
"Law & Order: Trial by Jury"
"The Beat"
"Law & Order"
"Homicide: Life on the Street"
"The X-Files"

Using him as the singular connection point, nine different television universes can be seen to be paired. The character John Munch also appeared on Sesame Street, however, not played by the same actor.

What about the Happy Days Universe?

• Happy Days

• Joanie Loves Chachi

• Laverne and Shirley

• Mork and Mindy

• Sabrina the Teenage Witch

If we disqualify the Tommy Westphall Universe…

An issue with the Tommy Westphall universe is that many shows are only linked by Easter eggs (e.g. the TARDIS is in the background of a single shot, things like Morley cigarettes or the Yoyodyne company exist), which isn't really in the spirit of the question.

Also, plenty of shows in that universe are not at all intended to be in continuity with each other. For instance, CSI: Miami as it's own show is not intended to be in continuity with either The Adventures of Superman or The Addams Family. So if we only count shows that have explicit, intentional links to all the other shows in the shared universe, the biggest is:

The Super Sentai franchise (41 Shows and Growing)

English speakers may be more familiar with Power Rangers, which takes fighting footage from this Japanese series and dubs it in English while splicing in original English footage. It began in 1975 with Himitsu Sentai Gorenger and now produces a new show every year, with the 2017-2018 show being Uchu Sentai Kyuranger.

The 35th show (Kaizoku Sentai Gokaiger) was an anniversary show that made it abundantly clear that it was in continuity with all the previous shows. The first episode had all the protagonists appear together (played by suit actors, which also happened in the original shows) and some of the non-suit actor original cast members reprised their roles over the course of the series.

Each subsequent show is also in continuity with the show before it, with the final shot of each show showing a hand-off between the Red Ranger of the current show and the Red Ranger of the next show. There is often a team-up episode between the current show and previous show as well, although nowadays, those are usually done in theatrical films. Speaking of which…

If we count movies, then the Super Sentai, Karen Rider, and Metal Heroes shared universe (84 Shows and Growing)

You said that movies don't count for this purpose, but if it did, there is the Japan-only theatrical movie Kamen Rider × Super Sentai × Space Sheriff: Super Hero Taisen Z, which is a crossover between the Super Sentai franchise, the Kamen Rider franchise (26 shows), and the Metal Heroes franchise (17 shows). Note that Super Sentai is represented by Kaizoku Sentai Gokaiger, the anniversary show listed above. So if movies were allowed, this would be a crossover universe with 84 shows at the time of this writing.

There were the Beverly Hillbillies/Green Acres/Petticoat Junction crossovers in the 1960s.

Buffy and Angel? Quite a few episodes there.

If you’re looking for a television crossover universe with the greatest number of episodes, you should probably be looking at daytime television soap operates which have been on the air for decades and have aired thousands of episodes. For example As the World Turns (13,858 episodes) spun off Somerset (1710 episodes) and Our Private World (38 episodes). It has crossovers with characters from Another World (8891 episodes) and The Young and the Restless (10,692 episodes and counting) which has in turn crossovered with The Bold and the Beautiful (7103 episodes and counting). By my count that’s well over 33,000 episodes among these series – well more than all of the Star Trek series, Doctor Who and related spinoffs or the Tommy Westphal-connected series.

Chicago Fire and Chicago PD Crossover and Law & Order Arrow and The Flash The Vampire Dairies and The Originals