Donald E. Westlake wrote a series of books around the adventures of a hapless thief named John Dortmunder (resulting in several movies, including The Hot Rock) and, under the pseudonym Richard Stark, a more cold-blooded series revolving around a professional thief named Parker (more movies, including Point Blank, starring Lee Marvin)
What’s Latin for D’oh?
Bryant and May series (mysteries with a splash of the weird) by Christopher Fowler. The adventures of two golden age detectives in modern London – in other words, they are relics, but still functioning, barely. Great reading, but maybe not one right after the other, at least not for me. Some of them get a little repetitive in tone, although not at all in plot. I highly recommend them. I don’t know how many there are, currently I am on #16, but they can be read in any order.
The Thursday Next novels by Jasper Fforde.
The five “Demon Prince” novels by Jack Vance. They aren’t fantasy, as the title might suggest, but far-future SF, where one guy tracks down five master villains who killed his family. One really nice feature of these books is that, while they do have an internal chronology, you don’t have to read 'em in order. You can start with any of them and go on to the others, any which way.
Vance also wrote a four book series, “Tschai, Planet of Adventure,” which is somewhat reminiscent of John Carter of Mars. These, you need to read in order.
Both series are just plain fun, some of the best “space opera” ever.
You already mentioned J.R.R. Tolkien’s The Lord of the Rings, so I’ll also recommend:
A Song of Ice and Fire by George R.R. Martin, about a power struggle in a fictional quasi-medieval kingdom. He hasn’t finished the series yet, alas, but what he’s done is big, intriguing and well worth a read. Start with A Game of Thrones.
The Spenser series by Robert B. Parker, about a smartass private eye in Boston. Clever, often exciting and always good fun. Start with The Godwulf Manuscript.
The Aubrey-Maturin series by Patrick O’Brian, about a Royal Navy captain and his ship’s surgeon/naturalist/spy best friend, set against the backdrop of the Napoleonic Wars. Exciting, immersive, richly detailed and often funny. Start with Master and Commander.
…as an example of what he wasn’t looking for. Are the Song of Ice and Fire books standalones? (Honest question; I haven’t read them.)
I would not consider them so. IMO. Not IMHO because about reading I am not humble.
The Bernie Gunther series by Philip Kerr. The way the books are structured you kind of get two stories in each book - a postwar story accompanied by related WWII or prewar flashbacks. Sadly, Kerr died recently but he left behind 14 consistently entertaining detective/suspense novels.
The Mr. and Mrs. North mysteries by Richard and Frances Lockridge. Written from the 1940s to the 1960s.
Fowler says that some are better read out of order, but cautions that On The Loose and Off The Rails should be read together and in order.
What about Michael Connelly’s Bosch books, which branch out to draw in other series.
Haven’t read a bad one of those yet.
I’ve been reading Tony Dunbar’s Tubby Dubonnet series.
Tubby is a New Orleans lawyer who struggles to hold onto his moral compass while getting mixed up in all sorts of misadventures.
I find the novels to be fun and fairly quick reads. The imagery and sense of place are very rich, with the city of New Orleans playing a major role throughout.
How about Robert Aspirin’s Phule’s Company series? While some of the characters and situations have consequences that come back around in a later book, each book tells a story by itself.
I wish Netflix would buy into this and make it a TV show.
Brian Freeman’s Jonathan Stride series have all been really good
Thank you all for the suggestions 
!
Piers Anthony’s Xanth books? There’s a million of them and each one is pretty much a standalone story but they revolve around the same group of character with each one having a different set of protagonists from that group.
There’s been a lot of controversy about pedophilia and misogyny in Piers Anthony’s books. You can google it, and it’s also been discussed on a few threads on the SDMB.
Did you skip Black ice?
I thought that was pretty bad, like a weak episode of a TV crime show. But it was only the second one. He got better.
I thought there was just one Spenser book, and he just kept writing it? “Who kidnaps Susan this book?”
(and I like Spenser!)