There are shows that have running shows in them, the obvious one is Itchy and Scratchy from The Simpsons. The other one I can think of is Monsignor Martinez (not sure if there's a name for the show or if that's the name) that's from King of the Hill.

Can you think of any others?

The Simpsons:
I on Springfield
Police Cops

Married...with Children:
Psycho Dad

Arrested Development:
Girls with Low Self Esteem

Futurama:
All My Circuits
The Scary Door
Single Female Lawyer
Everyone Loves Hypnotoad

Twin Peaks: Invitation to Love

I'm sure there's a ton on 30 Rock: TGS itself, MILF Island, etc...

If you really want to go down a rabbit hole on this subject, try this site: https://nestflix.fun/
It's a site made to look like Netflix, but all just show within show references.

.....Estrada or Nada from My Name is Earl

The Adventures of Captain Proton from Star Trek: Voyager.

Count Floyd's Scary Stories from The Completely Mental Misadventures of Ed Grimley.

Big Bang Theory and Young Sheldon: Professor Proton

Jerry on Seinfeld.

Featured prominently in The Dick Van Dyke Show: The Alan Brady Show.

Johnny Karate

From Daria:

There's probably a whole subcategory of shows where the characters work on a fake TV show. Even more if you include TV news shows.

30 Rock, Dick Van Dyke, Studio 60, On the Air, Back to You, Buffalo Bill, The Newsroom, Sports Night, The Mary Tyler Moore Show, Kenan and tons more.

...The Larry Sanders Show (The Larry Sanders Show), Extras (When the Whistle Blows), Home Improvement (Tool Time), The Joe Schmo Show (Lap of Luxury, Last Chance for Love, Full Bounty), Arrested Development (Scandalmakers, Mock Trial with J. Reinhold feat William Hung & His Hung Jury), Corporate (Society Tomorrow), SCTV (many, including Mailbag, Critic's Corner, Starting Out, Wrong Side of the Bed, Theater Beat, Shoot for the Stars, Stand Up and Be Counted, and TV Talk)...

I liked how The O.C. poked fun at itself with the creation of The Valley, basically an onscreen parody of itself. But then the O.C. also inspired the creation of the actual reality shows Laguna Beach: The Real Orange County and The Hills. Which in turn inspired the O.C. to create Sherman Oaks: The Real Valley, which is both a fake reality version of the fake television show it created and is also poking fun at Laguna Beach. Very meta.

Oh, and to correct the OP, Itchy and Scratchy is technically a show-within-a-show-within-a-show, as it airs on the Krusty the Klown Show*

* with the exception of the time it briefly moved to The Gabbo Show.

'TV Funhouse' on SNL is kinda a show-within-a-show.

Friends (Days of Our Lives, Mac & C.H.E.E.S.E.)

Most SNL skits are fake TV shows

Yeah. What's up with that?

Doesn't fit the topic but, I can't help thinking of the Batman episode where Batman and Robin are watching an episode of The Green Hornet and Robin says, "Gee Batman, they sure look stupid in those costumes,"

On "Entourage," there was a show called "Johnny Bannans" that one of the b list members stared in. Just popped into my head as I heard a line on a TV show about someone going bannans.

Which Batman series? Because I seem to be remembering that the old Adam West Batman had them actually meet The Green Hornet and Kato, right?

Yes, they met them in a team-up episode. Another episode has Batman and Robin watching Green Hornet TV show. A Green Hornet episode has somebody watching Batman TV show. I think it's some criminals in their hideout.