I think the Season 1 of Play With Me Sesame shows the classic rhyming game skit many times. Here's how it went:
Bert: (who is reading the "Cooking with Oatmeal" book) You take four cups of lump...two beated mangoes...
Ernie: Hey, Bert!
Bert: ...buttermilk...
Ernie: (Continued) It's time for the rhyming game, Bert!
Bert: Oh, yeah. But I just want to read my book.
Ernie: Oh, you'll love this game, Bert. You see, when you say something, I will say something that rhymes with what you just said.
Bert: Uh-huh, right. But I don't want to play your game.
Ernie: Maybe not, Bert, but we'll do it all the same. See, both game and same rhyme.
Bert: But, Ernie, I have to do something what I want to do.
Ernie: Wiggle your finger and buckle your shoe!
(Drum music starts)
Bert: Uh, Ernie, I'd rather just sit here and read my book.
Ernie: With ears you hear and with eyes you look.
Bert: Oh, come on Ernie, cut this out.
Ernie: Easy ol' buddy, you don't need to shout.
Bert: I don't want to play the game of rhymes!
Ernie: Nice goin' Bert, come ring my chimes!
Bert: (who is putting the book down) I don't wanna do it!
Ernie: There's really nothing to it.
Bert: It's so silly.
Ernie: A boy named Billy!
Bert: Stop it, please!
Ernie: A dog has fleas!
Bert: Ow!
Ernie: Pow!
Bert: Ernie, please, this has got to stop.
Ernie: Some people like to jump and hop!
Bert: I don't wanna do it (Ernie still interrupts!) anymore.
Ernie: What hmm hmm?
Bert: What hmm hmm?!? I don't wanna do it anymore!
Ernie: Oh. One and two and three and four.
Bert: Ernie!
Ernie: Bernie!
Bert: Stop it!
Ernie: Mop it!
Bert: Stop!
Ernie: Flop!
Bert: No!
Ernie: Flow!
Bert: Mmmmm....HIPPOPOTAMUS!!!
Ernie: Uh...ripotamus!
Bert: Ugh! All right! All right! I'd give up.
Ernie: Eat from the plate and drink from the cup.
Bert: I'll just have to play along.
Ernie: It plays like a game and sings like a song.
Bert: You're maybe right, I'm maybe wrong.
Ernie: Clocks go tick and bells go bong.
Bert: Hey, this game's kinda kicky!
Ernie: Sugar's sweet and glue is sticky!
Bert: Hey, the weather is wet and dust is dry!
Ernie: Dogs gotta walk and birds gotta fly.
Bert: It's the great game you've played with a friend!
Ernie: And now, Bert ol' buddy, we've come to the end!
(Drum music stops)
Bert: Yeah! You're right, Ernie, that's a great game! You-(still in rhythm)
Ernie: Uh, well Bert, I think I'm gonna read a book right now. Mmm hmm mmm...(Leaves)
(A moment of silence. We see Bert staring at the lamp.)
Bert: Hey there, lamp! That's a nice shade! You...it's not the same. (Leaves also)