You know how like The Big Bang Theory was spunoff into Young Sheldon, which has now been spunoff into Georgie & Mandy’s First Marriage (3 points!)? Which show has the most? For example, I’m pretty sure that All in the Family inspired at least two spinoffs (The Jeffersons and Good Times(?), one of which inspired at least one other spinoff) … is that the winner?
Obviously the Law & Order franchise and/or the CSI franchise is going to win, but what about other shows, sitcoms in particular?
There’s Maude, too.
If you’re talking sitcoms only: It might be All in the Family (with the four already mentioned). The Cosby Show had at least two; The Andy Griffith Show had two also, I think.
The Mary Tyler Moore Show had Rhoda and Phyllis, and then a non-sitcom drama, Lou Grant.
But if you’re not limiting it to sitcoms, I’d think the original Star Trek would be the winner.
I think it is All in the Family with:
Maude,
The Jeffersons,
Good Times,
Archie’s Place
All in the Family had Archie Bunker’s Place, Gloria, The Jeffersons, Maude, and 704 Hauser spun off directly from the originating series. Maude, in turn, spun off Good Times and The Jeffersons spun off Checking In. So that’s eight.
eta:ninja’d
The winner appears to be Law & Order:
Would it be possible to ban all mention of Tommy Westphall and his universe from this thread?
If you’re including the Toronto show (basically a Canadian version), you should also include Law & Order: UK and Wikipedia also mentions a French adaptation. Though I think we should exclude foreign adaptations.
Star Trek TOS
Star Trek TAS
Star Trek TNG
Star Trek DS9
Star Trek VOY
Star Trek DIS
Star Trek PIC
Star Trek TMP
Star Trek II
Star Trek III
Star Trek IV
Star Trek V
Star Trek VI
Star Trek Generations
Star Trek First Contact
Star Trek Insurrection
Star Trek
Star Trek Nemesis
Star Trek Into Darkness
Star Trek Beyond
Have I missed any?
Wikipedia mentions seven spinoffs of Happy Days.
I only remember Laverne & Shirley, Mork & Mindy, and Joanie Loves Chachi, but those three sprang easily to mind.
I don’t think movies should be counted. But you missed several Star Trek series. The question will be if streaming series count, and I suspect they will.
You missed:
With those, I guess the franchise has passed L&O.
Mork and Mindy wasn’t really a spin off, it was just using Happy Days as a launch pad for a completely unrelated show. But if you go with that, all those series could be a spin-off from Love American Style.
The OP asked for “shows” without specifying TV series. I guess I still consider movies “shows” too.
Enterprise I’ve seen, definitely should have been on the list. The others you mention I’ve either not seen or have never even heard of.
To explain to non-fans, the full names follow:
Star Trek: The Old Show
Star Trek: Time and Space
Star Trek: The New Guys
Star Trek: Danger - Speedlimit: 999
Star Trek: Vulcans or Yoda
Star Trek: Dogs in Space
Star Trek: Planets in Confusion
Star Trek: Those Muppet Pigs
Those are actually pretty good! 
They’re on the roadmap for Phase(r) 3.
Doctor Who (the original series)
Doctor Who (the revived series)
“Doctor Who and the Daleks” (movie)
“Daleks’ Invasion Earth 2150 A.D” (movie)
“Doctor Who” (the TV movie)
“The Sarah Jane Adventures” (spinoff series)
“K9 and Company” (spinoff series)
“Class” (spinoff series)
“The War Between the Land and the Sea” (upcoming spinoff series)
“An Adventure in Time and Space” (TV movie about the origin of the series)
“Doctor Who Confidential” (documentary series akin to “Talking Dead”)
“Sarah Jane’s Alien Files” (another spinoff series)
“K9” (animated spinoff series)
" Daleks!" (animated spinoff series)
(and there may be more)
Anybody mentioned JAG yet?
Those Old Scientists.
I think Star Trek wins, with L&O second.