All of the following air on Radio Four. Some of them can be listened to on the iPlayer, if they are airing currently, some can be found on YouTube if you're lucky, and I think all can be found to buy from iTunes or Audible. I'm sure there are less legal ways, as well, if you are so inclined to seek them out.
Some of these have been on for decades, so don't be alarmed by the number of episodes. And these are just my favourites, or those I thought people might enjoy. There's a lot more out there and I highly recommend looking for it.
Elvenquest (4 Series, 24 episodes)
Summary: Sam is an author of fantasy books who is kidnapped by a group of questers from another world, who are in search of the Chosen One to help them find the Sword of Aznagar and defeat Lord Darkness. The Chosen One is Sam's dog, Amos, who is transformed into a human in the new world.
Pros and Cons: If you are a fan of the fantasy genre of books or MMORPGS, you'll likely enjoy the send up of a Lord of the Rings-type quest. The humour is largely based on how ridiculous the set up to most fantasy novels are. It's not a laugh a minute, and there's a lot of sexist humour (self-aware sexist humour, it's humourous because it's aware how sexist it is, if that makes sense), which might not be everyone's cup of tea. I've heard Series One, which was repeated last autumn, and Series Four, which is currently airing. Both are solid and enjoyable.
Warhorses of Letters (2 Series, Eight Episodes)
Summary: A series of correspondance between Napolean's horse, the stately Marengo, and Wellington's Horse, the youthful Copenhagen. They are in love, but torn apart by war.
Pros and Cons: Marengo is voiced by Stephen Fry, who is always delightful. The humour is good, if you like horse puns. The episodes are only fifteen minutes each, which can be a good or bad thing depending on how much entertainment you're looking for.
Bleak Expectations (5 Series, 30 Episodes)
Summary: This one is a bit hard to explain without going on at length. It's basically a complete send up of all the tropes in a Dickens work. Pip Bin is an orphan, who is sent to St. Bastard's School by his guardian Mr. Gently-Benevolent, where he meets Harry Biscuit, his best friend. Their adventures are ridculous and convoluted.
Pros and Cons: If you know Dickens, it is quite entertaining. For a while, at least. I lost interest around Series Four, as the ridiculous got a bit too ridiculous, and the plots were largely repetitive. The first few series are a lot of fun, though, and the voice actors are wonderful.
The Unbelievable Truth (10 Series, 60 Episodes)
Summary: Panelists give a short lecture on a given subject, which is entirely false, except for five hidden truths. The other panelists have to spot the truths.
Pros and Cons: I don't think there's been an episode I haven't enjoyed. The lectures are silly and fun, the facts are interesting, and the banter is always spot on. David Mitchell hosts and gets his rant on frequently.
Just a Minute (65 series, 796 Episodes)
Summary: Panelists are asked to speak on a topic for a minute, without repeating anything but the words in the topic, deviating from the topic, or hesitation while speaking. It's harder than it sounds. Panelists ring in if the person speaking makes a mistake, and then they inherit the topic and keep speaking until someone rings in on them. Whoever is speaking at the end of a minute (counted only when people are speaking on the topic), gets a point.
Pros and Cons: This one is hit or miss, depending on the guests. They've been getting some young people in lately, which is helping to mix things up. It isn't my very favourite, but it's good background noise with some solid laughs.
I'm Sorry I Haven't a Clue (58 Series, 420 Episodes)
Summary: The self-described 'antidote to panel games', it is a collection of puns and singing and games that make no sense.
Pros and Cons: I really, really love this one. It is very silly, but I genuinely laugh along to most of it. The games repeat and the humour repeats (there are a lot of established jokes that it takes a new listener a while to pick up), but there's always a lot of fun.
The Horne Section (2 Series, 12 episodes)
Summary: Musical comedy from Alex Horne and his band.
Pros and Cons: I find this one inexplicable delightful. The humour is lovely, the music is catchy, and I find it just makes me happy.
John Finnemore's Souvenir Programme (2 Series, 11 Episodes)
Summary: Sketch comedy written by and starring John Finnemore.
Pros and Cons: Probably the best Cabin Pressure void-filler you'll find. It's highly entertaining, the sketches are, as a rule, hilarious, and the humour matches up with Cabin Pressure very well.