Franchise Killer: Aside from appearing and making cameos in other games, The last official game was Jak and Daxter: The Lost Frontier due to being the lowest in reviews and sales despite plans for a fourth installment. With the main series of games being exclusive to the PS2 era excluding spin-offs with PS3 and PS4 HD Rereleases.
Follow the Leader: It is by no mere coincidence that Jak II was made considerably darker than its predecessor and was released around the time people would just be finishing up Vice City and needing something to placate their sandboxing fix. The first game is also a blatant Super Mario 64 clone, but it's more forgivable than most due to the game's absolutely exquisite presentation. The transfer from "traditional collectathon platformer" to "action game with some platforming elements" is generally considered justified, as open world steal-car games went from strength to strength in the early 2000s, while character platformers nearly disappeared entirely for the next decade (unless they starred Mario).
God Never Said That: It was thought that there was going to be a third main character for the first game but this was later dismissed by Andy Gavin in a later interview.
Old Shame: The first game seems to be mostly ignored by Naughty Dog its self only mentioning the sequels but does get ports and remasters for later consoles. Despite being the best-selling game in the series the game didn't resonate with the global market like how Crash Bandicoot did.
Troubled Production: When the first game started production, the developers were so prioritized with the game's technology (Working on it for almost 2 years), they didn't get to the gameplay coding until fairly late in production. They had to design settings and enemies for the game not knowing if they would work properly, but according to programmer Steven White:
Early designs of Daxter were fox like, then made as a monkey, then finally as a otter/weasel hybrid named as an ottsel.
In the Jak X commentary, they say that the original plan was to make Rayn "more Rubenesque" to (sort of) match her father. They decided to go for the "standard" Naughty Dog female model for her, instead.
Also according to the Jak 3 Design Bible Rayn would have appeared earlier in the series, Being the former boss of Jinx and the semi-ex-girlfriend of Daxter, Possibly appearing in the spin-off game Daxter.
Given the implications at the end of the first game, it's not hard to think that Jak II might have been at least considered for a more lighthearted tone like the first game. Of course, if there were any such plans, they were abandoned in favor of a Darker and Edgier sequel that changed a lot.
The Lost Frontier was conceived and started by Naughty Dog. It was originally going to be finished by them as well, but difficulties with the Uncharted project on PS3 (and maybe other problems as well) caused Lost Frontier to be cancelled early in development, and all remaining developers migrate to Uncharted. The final version of the game is a recreation instead of a port, based somewhat on the original script and gameplay ideas, but not what the first planned version of the game would have been. For example, Dark Daxter, despite being an idea kicked around during Jak 3's pre-production, was never meant to be in TLF in any form.