I think Rincewind is so interesting because he never learns any impactful lesson, but he's still such a relatable character. He's got these flashes of deeper character that could almost be missed beneath the rest of the story, and these little flashes always made me empathize with him more.
There's a line in CoM where Rincewind is described as liking Twoflower because "disliking him would be like kicking a puppy." It's hinted they have more interactions off-screen that further their friendship, like the parallel dimension in which their Roundworld counterparts become friends at an airport and sit together on the plane. Throughout TLF Twoflower's revealed to have more insight into Rincewind than expected from his naive character, quietly pointing out to other characters that he knows Rincewind is no good at magic and getting fed up with Rincewind toward the end by cutting off a sarcastic remark before Rincewind can make it. All throughout they have dialogue and camaraderie, to the point that Rincewind actually plans for Twoflower to stay at UU at the end because in Rincewind's mind, of course Twoflower would be sticking around, making his disappointment all the more palpable when Twoflower reveals he's going home. I teared up at that scene as a kid and I do it now whenever I re-read it. His care for Twoflower carries across the series, shown when he begins reminiscing about Twoflower in IT long before the two are reunited, showing Twoflower left more of an impact on Rincewind beyond being the naive tourist.
His interactions with the Librarian show this side of his character, too. At the end of Sourcery, the Librarian is the one grieving over Rincewind disappearing into the Dungeon Dimensions, holding onto his hat until IT, and jumping at the chance of him being brought back. Sourcery shows their friendship and hints more at Rincewind's past as a library assistant through Rincewind understanding the Librarian, showing their was development between LF and Sourcery. I find this friendship more meaningful because we've seen how the librarian treats intruders in the library, yet he formed this bond with Rincewind, despite the fact that Rincewind being guilty of that is the major plot point for the first two books. Moreover, Rincewind is the only one on the staff who even knows the Librarian's name. Not only is Rincewind close with him, he's likely the closest to the librarian in the entire city.
I argue Rincewind has more passion than most other characters. He doesn't just like Ankh-Morpork, he loves Ankh-Morpork. He avoids danger and heroism to the extreme except when it involves going home. The only reason he enters the traveling shop in LF is because it can get him back to Ankh. He faces Coin with a brick in a sock in Sourcery just to defend his home. In IT, he was perfectly happy on the island at the beginning, but cries with joy at being back in Ankh, eating Dibbler's sausages and embracing a beating from the Thieves Guild. He identifies so intensely as a wizzard and citizen of Ankh that he agrees (albeit under duress) to a world-saving adventure in the Agatean Empire just to be allowed to hold the title and live in UU. He's a survivalist in that he'll adapt to wherever he ends up (the island in IT, Fourecks in TLC), but when the idea of returning to Ankh presents itself, he's reluctant, terrified, and begrudging, but ultimately willing.
An aspect of Rincewind's character that goes overlooked is his fascination with technology and "science" as it's present in the Disc. He first shows this in CoM with the iconograph. After using it, the book describes him as "taking every opportunity to operate the device," contemplating how it works, "freezing light particles" to capture an image. Later he talks about "harnessing the lightning" and bemoans the shortcomings of magic, thinking there must be something more practical, like science and mechanics. Considering the industrial revolution of Discworld didn't begin until several books later with MP, he not only demonstrates a fascination with new technologies, it puts him ahead of his time. His affinity for technology appears again in IT when he discovers the terracotta warriors. He immediately takes to controlling them, figuring out how they work almost instantly, and immediately begins applying it, even if only to make an entire army flip the bird. It's to a lesser extent, but even in TLC there's a scene where he gets thrown out of a meeting room because he keeps playing with the bullroarer he'd just found. After using it outside, he finds he can use it to make it rain. In his final appearance in Raising Steam, while he's hiding under the seat of the train, he still got on it, and in those footnotes he does admit to its practical uses. It's a small glimpse of Rincewind, but entirely consistent with his character in more ways than one.
Finally, Rincewind always gets the reputation as a failed wizzard. In actuality, he's a fantastic wizzard. In several instances in CoM he goes on about magic theory, how spellcasting works, and magic's history. He explains exactly how spells are cast and what their faults are. He knows the ins and outs of magic in incredible detail. He's an expert on wizzardy in every sense but the practical. He knows magic, he just can't perform it.
It's true Rincewind doesn't go through drastic character changes like Vimes and Granny Weatherwax, but the character he does have is still an enjoyable and relatable one. He's often remembered for his most prominent traits of being a snarky coward and the epitome of the reluctant hero trope. But he has so much more to his personality that slips through the lines and is easily missed. I love Rincewind because each of his books is a balance of slapstick, sarcastic humor, and brief, shining moments of pure humanity that get right to me every time. There is at least one point in every Rincewind novel that makes me cry for him, not out of sympathy, but empathy.