For non-starter , non-legendary Pokemon:
1.

Gengar
Was at an amusement park earlier today and I saw this guy's plushie right next to most of the other commonly giga-iconic Pokemon like Charizard, Pikachu, Squirtle, and Snorlax. This is particularly notable to me since I feel Gengar didn't really get special attention in generation 1 compared to the other Pokemon I listed.
Honorable mention goes to Snorlax, but I feel its always been a special Pokemon with its encounter method.
2.

Azumarill
Generation 2 is really hard to find a "breakout star from". I feel Azumarill is one of the Pokemon with the longest standing impact from generation 2 even though its difficult to obtain in the game itself. A good case could be made for ampharos since its a 3 stage line from the beginning, as well as a few other generally beloved gen 2 Pokemon like Scizor and Heracross, which I feel still have staying power to this day.
3.

Gardevoir
Yeah Gardevoir is the pretty obvious standout gem here. 3 Stage line that availible relatively early and has garnered a massive fanbase over time. Seeing its merchandise constantly sold out on pokemon.com and its appearance in secondary media sort of solidifies this.
4.

Lucario
Kinda cheating since Lucario was hyped up to the moon when gen 4 was releasing, with a smash appearance to boot. Its popularity has taken a big hit over time however, imo anyways. Nonetheless, we still see it get a lot of special attention in later titles.
5.

Chandelure
Despite its basic premise, Chandelure is an extremely popular Pokemon, getting a lot of appearances in side-games like Unite and Pokken, while also being quite memorable due to its great design and personality.
6.

Aegislash
Another cracked 3 staged line you get early in the game. Very popular Pokemon for many of the same reasons as Chandelure.
7.

Mimikyu
Extremely popular Pokemon due to its tragic backstory while also being part of the 20 anniversary celebration in some capacity.
8.

Dragapult
This Pokemon isn't really memorable to use in game, but has had many appearances in side games like Unite.
9.

Tinkaton
This Pokemon has an incredibly popular fandom since generation 9's launch due to its unique relationship with Corviknight and has only grown in popularity, being one of the first Pokemon to be featured in Pokemon mastered prominently. Tinkaton also has a notable role as an NPC in Pokemon unite.
Main thing you'll notice is that Ghost and Fairy types are generally the standouts every gen lol.