1) Fire Emblem
My favorite Nintendo franchise is Fire Emblem. One of the first games I can remember from my childhood is Shining Force: The Legacy of Great Intention, a turn-based tactical RPG for the Sega Genesis, since my brother played it and I'd watch sometimes. When I eventually learned about Fire Emblem when I was older, the turn-based tactical gameplay and the similarities (knights, mages, archers, countries at war, etc.) really appealed to me due to my nostalgia for Shining Force. My favorite video game of all time is Fire Emblem: Path of Radiance (and its sequel Fire Emblem: Radiant Dawn, since it takes place three years later and features the same setting, returning and new characters, etc.), so the series definitely gets a boost from that.
I have some things I don't like about the series (Awakening and Fates era increased emphasis on shipping, Tokyo Mirage Sessions, recent games having player avatars instead of letting the lords be the main lead character like the older games, etc.), but hey, it's whatever. Love the gameplay, love the characters and their conversations, love the settings, love the music.
I wish more people were interested in the series, and I also wish it had released in the West from the start back in 1990 instead of waiting until Fire Emblem 7 in 2003.  It feels undeservedly overlooked amid the Nintendo franchises even though it's older than Animal Crossing, F-Zero, Kirby, Pokémon, and Star Fox. Heck, it's older than Wario! Granted, turn-based tactical won't appeal to everyone, but I have to wonder whether it wouldn't feel so comparatively niche if it had international exposure back in the NES days like Mario, Zelda, Kid Icarus, Metroid, and the like enjoyed. Maybe there would be less people annoyed about blue-haired anime swordfighters in Smash if more people had experience with the games to know the characters, and hey, maybe there'd have been more variety in the character selection (and the weapons they use) from the series instead of sticking with the lords and avatar characters.
2) Pokémon
Pokémon is second. I've been playing the series since childhood, starting with the initial boom in the 90s when Red and Blue first released in the United States back when I was a kid. Peer pressure got me to quit during Gen III, but a friend got me back into the series when they mentioned they were playing HeartGold and SoulSilver (I loved Crystal) and I got my own copies soon thereafter. I got back into the series hard; I got copies of Ruby, Emerald, FireRed, Leaf Green, Diamond, Pearl, and Platinum since I'd missed their releases during my break, and I've gotten all of the (mainline) installments on release since Black & White. It's the series that I've spent the most time playing throughout my life. Battling is fun, breeding and team building is fun, I love hunting shinies with the Masuda Method.
Didn't really enjoy Sword/Shield for numerous reasons (Pokédex culling, raids feel poorly implemented and don't appeal to me, I find Dynamaxing to be a bad gimmick) and was prepared to step away from the series moving forward and stick to the earlier games, but Legends Arceus hooked me back in well enough to have me wanting to give Scarlet and Violet a shot in November.
3) EarthBound/Mother
I love this quirky little series. Quirky humor, lovable characters, great music, fun combat (and I love that ticking health mechanic). Nostalgic for me since EarthBound is another game I remember watching my brother play, and I inherited the game since he stopped playing games as he got older. I got a couple friends into the series by talking about it and they fell in love with it too. EarthBound was great on its own, but Mother 3 is absolutely magnificent and . I still need to get around to EarthBound Origins to fully appreciate the series, I never have due to it seeming a bit archaic and since I felt like I knew the gist of it from reading about it. Even still, this series has a special place in my heart.
4) Super Smash Bros.
I've always been into fighting games. I never really played platformers like Mario or Metroid when I was young, so I'm pretty godawful at them. Comparatively, I've been exposed to fighting games since I was about 2 or 3 since my brother had Mortal Kombat and Street Fighter II. As such, I'm more comfortable with such games. Super Smash Bros. and games it inspired are their own sort of beast with their focus on percentage damage and making your opponent lose stocks by ring out self destructs, but hey, it's close enough. Punching, kicking, jumping, crouching, throwing, blocking special moves, all that good stuff. Been playing the series since the first one on the N64 and it's been a blast.
5) Animal Crossing
Well, I'm here on an Animal Crossing forum, so it's to be expected I guess. Saw a friend playing it when I went to visit them one day back in 2002, got myself a copy so we could play together, and been hooked since. Didn't know about Wild World at the time, disliked City Folk, but absolutely adore New Leaf and have some qualms with but still really enjoy New Horizons. The animals are fun to socialize with, the NPCs have great personalities, and I love designing homes.  Delightful little world.
Honorable mentions to:
Luigi's Mansion: I couldn't get into Dark Moon despite trying, and I've never bothered to try Luigi's Mansion 3 yet, but I absolutely adore the original Luigi's Mansion. I really love the general concept as well. I wish they'd stuck with the humanoid style of ghosts they had in the original though.
Mario Party: The series has had its ups and downs, but when it's great, it's great. I'm most partial to 2 and its selection of minigames, and I'm glad Horror Land and Space Land boards got to come back in the Switch's Mario Party Superstars. Mario Party makes for great fun to have with some friends. Sometimes strife and turmoil too, but you know that going in.
Metroid: I've never played any of them to be honest, but I love Super Metroid. I've watched people play it over the years during AGDQ/SGDQ speedrunning charity marathons, and I really love the designs, the setting and the atmosphere. Samus seems pretty cool too!