There was a post on this site about stuck scenes in the movies. It was a great post! It got me to thinking back over the last 40 years of being a stucker myself about famous & noteworthy men that have been stuck in real life!
Most of my knowledge is from reading books& articles about various stuck experiences but a few include remembering seeing a tv clip or tv interview.
If you know of any not mentioned here, feel free to jump in and post what you know!
FYI- for the additional boot and smoke fetish guys, I'll post if I knew the guy was a smoker and/or wore boots!

Richard Widmark: Per an article on Richard Widmark's early life and career, the article stated that while borrowing his father's car to go to a college play (c. 1935-1940) he was in, he got his father's car stuck in the mud. The article said he was able to get it out of the mud and make it to his play on time.
(Smoker and wore western boots in his western film roles)

John Travolta: Per a Good Morning/Rona Barrett Hollywood tv segment in the early 1980's and after he'd made SNF and Urban Cowboy, Rona Barrett reported that Travolta and a group of friends were vacationing in Colorado and that the SUV that Travolta was driving became stuck in a snow drift. From what I remember her saying, he and others were able to eventually get out the SUV (spinning the tires/pushing) and continue on their trip. I do remember her saying that Travolta was driving the SUV that got stuck as there were 2-3 total SUVS's on the trip since it was a group of friends with Travolta.
(Smoker? / definitely wears cowboy boots per "Urban Cowboy"- very sexy wing-tip cowboy boots, I might add! Other pics of him in cowboy boots and other style of boots!)

Clark Gable: Perhaps the most prolific "stuck" personality! In the book "Gable" by Jean Garceau, it is told that in Feb 1940 (after making GWTW) that for Gable's birthday, he and his wife along with another Hollywood power couple, drove into Mexico to go camping/shooting. While driving up into the mountains a heavy rainstorm occurred and Gable (driving) and his vehicle became STUCK IN THE MUD on a narrow county road! The story went on to say he and the husband of the other power couple tried to get the vehicle out- but, couldn't and spent the night in the stuck vehicle! They were pulled out the next day by a Mexican travelling the road!
Gable also had a snow stuck in his 1948 movie "Homecoming". About 3/4 th's through the movie, Gable driving a jeep...becomes stuck in snow/ice! The scene shows the tires spinning on the jeep!
As a side note, it's been written that Gable worked in the oil fields of Oklahoma & Texas in the 1920's. Oil fields are VERY muddy when it rains! So, I'm guessing that Mr. Gable knew very well the art of driving and getting STUCK!
(Smoker and boot wearer- especially, cowboy boots as he owned a ranch for many years and made several western films!)

Glenn Beck: In 2014 while on a family outing, Glenn Beck got his truck stuck in the mud in a field. There are pictures on the internet and he talks about it in a video blog during that time frame (Youtube).
(Cigar smoker/ work boots)

David Lettermen: Well known 2009 incident per the internet where David Lettermen got his truck stuck in mud on a muddy road in Montana where he has/had a residence while driving to the small Montana town for his wedding date! The article says that Lettermen and his future wife were driven into the small Montana town to be married- leaving his truck STUCK in the mud! Oh, Dave!
(Cigar smoker/ cowboy boots)

Jay Leno: Article I read where Jay was recanting his late teens/early 20's years. The article was brief - but, Leno did state that while on a double date, he got his vehicle stuck in the mud! The article didn't elaborate or say more.
Spin those tires, Jay!
(Boots)

President Bill Clinton: Per his autobiography, Bill Clinton recounts that during his high school driving years in Hot Springs, Arkansas, he went on a date. While leaving the parking area of where ever they were, Clinton got his car stuck in the mud on the date (c. 1962-1964). In the book, Clinton recalls the incident well and talks about how he spun his tires a lot to get out (till he smoked them?) and that the rear tires were "smooth as a baby's butt"!
(Cigar smoker/ cowboy boots)

Steve Irwin: In several of Steve Irwin's tv episodes "The Crocodile Hunter", he got his SUV stuck in both mud and sand due to his various filming locations! He was very verbal and would make comments while filming of "I'm getting bogged down!" "I'm STUCK", etc! Sexy guy when he got stuck doing his nature thing! ( Work/nature boots)

John F Kennedy, Jr: In the 2006 book written by a lifelong friend of JFK Jr (William Noonan), he recounts his long friendship with JFK Jr and their Hyannis Port vacations in their late teens and early 20's. The book make reference to the beach in Hyannis Port and reference to he and JFK Jr getting their jeep STUCK IN SAND (more than once) while driving on he beach!
(Smoker)

Will finish up in Part 2 (a short Part 2)