I like "Wait, Wait, Don't Tell" on NPR.
[Swears at Computer - I had completed my answer then inadvertently pressed a secret key combination I've never identified, but which deletes it all! Ho-hum. Start again.]
My favourite game shows, all on BBC Radio Four:
The News Quiz - satirises the real news and current affairs. I like too The NOW! Show!, similar satire but stand-up comedy.
I'm Sorry I Haven't A Clue bills itself as "the antidote to panel games" of which indeed it is a splendid spoof in which "the team are given silly things to do by the Chairman" and the [fictitious] Samantha does the scoring. Yes, it's ripe with risqué humour. Apparently the BBC sometimes receives requests for the rules to the show's totally-made-up London street-names game, 'Mornington Crescent', from baffled listeners who have failed to grasp that actually, there is no such game in reality.
Gentler comedy comes from the four contestants in Just A Minute, each trying to "speak without hesitation, deviation or repetition", but losing play to whoever is first spot such a fault and buzz, gaining a point if the challenge is correct within the rules. The subjects are given only at the round's start, and are totally inconsequential: one might ask for a thesis on "My favourite dessert" and the next be about "Polishing shoes" or something equally trite! It's a lot harder than you think, even for the regular players, and complete 60-second talks without interruption are fairly unusual.
Non-comedy, in fact more erudite but still entertaining are two quizzes:
There is a straightforward Q&A game ranging over music of all styles - sorry, but its name escapes me!
For real bafflement though, try Round Britain Quiz, played each time by two teams of two in different places. It's a warm, gentle but extraordinarily cerebral game whose entertainment value to we mere mortals listening is as much from the detective work by the players as from testing your own and gaining more knowledge, as you would in a simple Q&A quiz.
RBQ's questions typically link three or four obscure, seemingly unconnected references, and you need to identify them and the common thread. It helps to have a wide general and literary knowledge, and high skills in solving lateral-thinking and cryptic-crossword puzzles, as the methods are similar. I've occasionally twigged some of the easier items, but only rarely!
I'll give you a flavour, in a question in RBQ style but without its polish and probably a lot simpler. I won't give its five answers here yet, so you've a chance to try it!
"How have an elderly but persistent New World native, his aged parent in the Old and their allegedly naïve African and somewhat liverish Oriental cousins, kept trading companies afloat over the years?"