Leadville 100 podcast by Elden “Fat Cyclist” Nelson.
They had a fine first season with bonus episodes. @Jonathan was a regular guest there.
An off topic podcast: “You are not so smart” about various things how our minds play games with us. I recommend episodes about The Backfire Effect.
Specifically about mental training, the Consummate Athlete podcast has had a few recent episodes with the Brave Athlete authors about their book and methods. The authors were also recently on the Faster by Flo podcast.
I’m a “heavy” podcast listener… In addition to Trainerroad, I listen to:
MTB Podcast - Jolanda Neff’s BF and some Yeti fanboy host this podcast which I heard about when I was in Sweden.
Paceline - not as much as I used to since Fatty flew the coop, but it is still good and maybe getting better, Patrick and Selene are great journalists
Velonews, specifically FastTalk episodes (TechTalk is good too)
Cyclingtips - it’s fun, NerdAlerts can be especially informative
SingleTracks Mountain Bike Podcast - the recent interviews with Tom Ritchey, Joe Breeze, Charlie Kelly and Gary Fisher are great if you’re into MTB
*Sonya Looney - a lot depends on the guest and particular topic, but some are great
Faster by Flo - similar to Fastalk/TR but hosted by the Flo guys, also more guest centered. I wish the guys would push back more on their guests if the guest is saying something controversial.
Consummate Athlete - great guests a lot of the time, give Pete and Molly a chance - Pete’s a great MTBer and Molly seems to have good connections in the bike industry through her publications
BikeShop CX - Car Chat meets bikeshop - they make installing tubulars sound easy, but you know they’re lying.
FasCat Coaching - Frank Overton’s shop, aka Phil Gaimon’s coach and one of the first big promoters of SweetSpot
Gravel Lot - gravel/adventure riding oriented. I haven’t given it too many listens, but on the first bit it looks good.
Endurance Planet - I used to listen to this more, but they sell some waaay left field type information (call it a Ben Greenfield hangover). I honestly wouldn’t recommend it until I saw someone else did here.
Non-Cycling - Ezra Klein Show, Milk Street Kitchen, Eater Upsell, Modernist Bread Crumbs, Cooking Issues with Dave Arnold, SeriousEats, Sporkful, Meateater, The Daily, the Splendid Table
Lots of great suggestions here.
I’ll add a few:
Song Exploder (musicians take apart their songs, piece by piece)
StarTalk Radio (Neil Degrasse Tyson. Need I say more?)
ZwiftCast
Meb Faber Show (if you’re into investing)
In Our Time (BBC podcast on the history of ideas)
FiveThirtyEight Politics (US politics podcast. Very depressing recently)
Flo Cycling Podcast
EconTalk: Weekly interview-based podcast. Not necessarily about econ.
Dan Carlin’s Hardcore History
Ask a Cycling Coach - Hidden gem. Not sure if you’ve heard of this one?
Maybe too much! Thanks
Here’s my recommendations. I haven’t listened to everything as yet as there’s so much and some I’ve recently discovered. But my top 3 are there.
Ask a Cycling Coach - my fav podcast
Velonews - for the Fast talk episodes. My second fav podcast
Faster - Podcast by FLO. My third fav podcast
Cycling Time Trial Podcast - really good time trialling stuff
The Cycling Podcast - for World Tour road racing news
Medicine of Cycling - there’s audio and video versions
Others that are useful - I’ll listen to selective episodes or segments
FastCat Cycling Training Tips
Tailwind Coaching
TrainingPeaks Coach Cast
That Triathlon Show
CyclingTips
Rdella training
Bike Shop Show - it’s not current but great if you’re a home bike mechanic. Now it’s transitioned to Bike Shop CX which isn’t as good
Zwiftcast
A particular Episode from Flo’s Faster, with good ol’ Nate’s LVT100 as the focus.
https://www.podbean.com/site/EpisodeDownload/DIR4D183E8TS7SW
A couple of times they’ve mentioned on the podcast about a specific one to avoid describing it as a snake oil peddler what is it?
Cycling
Obviously Get Faster!
Cycling Tips - agree this has improved a lot recently
Cycling Time Trial Podcast
Triathlon
IM Talk
The Brick Session
Purple Patch
Fitter Radio
Other Sports Stuff
Tower 26 (Swimming related)
Endurance Planet occasionally if one of the topics interests me
Fitness Behavior Podacast - Bevan from IM Talk and is usually more self development/attitude type stuff and can be pretty good
Freakonomics
Stephen Dubner is a great host.
The Curbsiders
This one is medicine focused, specifically internal medicine. These are young guys who ask really thoughtful questions of experts, helps keep those of us in practice on top of recent advances.
GainCast
Science of Ultra
Embedded
Fresh Air
Pod Save America
To add a few.
I agree.
Are there any good podcasts about nutrition and clean eating for athletes?
Oh man. Don’t get me onto more podcasts. I’ve been listening to podcasts since 2004 but my library has been expanding a lot recently. I am subscribed to 22 at the moment. It feels like a full-time job trying to keep up (and I am going to be fired soon at the rate I am slipping behind).
Lots of good podcasts mentioned. A lot of podcasts simply described as “good” etc. It would be nice to mention what angle the podcast takes. Eg training, tech, race news etc. Sometimes the names of the podcast’s doesn’t make it obvious. As the OP was asking for mental toughness podcasts, what are training podcasts people subscribe to?
Thanks.
Fat- Burning Man with Able James is really good.
Couldn’ see a mention of “The Bike Show” by Jack Thurston upthread.
It’s a look at cycling from every angle. He is not regular but there is quite a back catalogue to keep you going. I never thought I would like it as a fan of pro road cycling but does a great interview with some great musical inserts. 
I would highly recommend the Tim Ferris podcast. He has a knack for interviewing people who have a really unique way of thinking or viewing the world. Even if the guest is from a field that you have no past interest in, there is always lots to take in.
The Joe Rogan podcast is also worth a listen as he has some interesting guests that provide really interesting discussions. I tend to skip the MMA/UFC themed stuff as that’s not really my thing but other than that it’s worth a listen.
I have listened to the first 5-6 episodes of FLO, great show, but it annoys me a little bit that they are interviewing guest through a lower quality telephone line.
Makes it a bit harder to listen to when driving my car (Solution - buy new car) 
Another Vote for The Cycling Podcast (and “Friend” subscription).
I also pay to subscribe to the Velocast (velocast.cc).
Flammecast
I also regularly listen to Cycling Tips and the Velonews podcasts, although I have found the constant pushing of periodisation meaning I’m less inclined to listen to it than I used to - I’ve taken a great deal from it in the past, I just think he’s going too all eggs in!
Food For Fitness and Sigma Nutrition for, eh, nutrition.
I occasionally listen to The Move. I don’t really warm to him though, but I think he’s right in the Tour Route one saying radios and Power meters shouldn’t be banned, but they should be open access!
Maybe a bit strange because this is not an English spoken podcast (Remko Zwanenburg brief — Live Slow Ride Fast). It is in Dutch so it applies only to the people who understand Dutch. But it is by Laurens ten Dam, a pro tour rider.