There are many anime that would be perfect for this, and it would likely depend on what the person likes. So I've made a recommendation by type instead of a one fits all type.
Fighting/Adventure Saint Seiya
This series is fairly old, but has a great plot, mythology, adventure & great character development that draws you in.
Fantasy. Fairy Tail
This one is a no brainer. Classic fantasy, magic and adventure all in one. The characters are great too.
Drama/Feels Train. Your Lie in April
The story, characters and music draw you in and captivate you. If you don't cry at this you're either dead or your heart is missing.
RomCom. My RomCom SNAFU or To Love Ru Both of these are good series with a decent plot, a some fan service and a variety of cliche characters.
Futuristic/ Worlds Colide. GATE This was one of those, meh I'll give it a shot, kind of shows. I'm glad I did because it has a good plot & the MC Is an otaku(bonus points). The plot didn't drag, animation was solid, it kept you interested and... Rory Mercury.
Futuristic Space based. Cowboy Bebop This series was absolutely awesome. The characters are solid with a good backstory, the action is well paced and the story never drug. The only thing I had a problem with was who was my favorite, Spike or Ed.
Mecha. Mobile Suit Gundam 00 I enjoyed this series, even in the dub version it was good. I have kinda followed the Gundam saga for a long time & it has never let me down.
Harem. Monster Musume I've seen a lot of harem shows, but this one stands out because of the interaction between the characters. Its also notable because almost all of the harem is monsters, the ecchi is not off the charts and they managed to even make the monster girls cute & likeable.
Trapped in a Video Game. Log Horizon I know everyone was expecting SAO, but I think Log Horizon did it better. The story is good, the characters vary, but unlike SAO, you don't get to give up and take an NPC slot, it requires interaction between the in game characters and the gamers. I liked the twist that you lose memories in exchange for reviving also.
Slice of Life. Love Live! I started this series because I was bored, I stayed for the characters, music, animation and story. This was a very pleasant surprise and I'm actually looking forward to the next season.
Mild Ecchi. Rosario Vampire The first season of this anime was enjoyable and didn't go overboard on the fan service. The story was also pretty good.
Hard Ecchi. Highschool DxD Its really hard to believe anyone watches this just for the story. That said it does have a good story, the characters are solid and likeable (mostly), and Issei has one of the most awesome fighting moves ever "Dress Break". If you need another reason to watch this it would be Rias &/or Ayano.
Slightly Twisted. KissxSis The typical sister x brother pairing, but cuter. Fortunately they are step- bro/sis, but it's cute, fun, and highlights several of the cultural differences between the US & Japan.
So there they are, if you haven't seen any of the above, give them a chance. I also will give 1 piece of advice... Do not watch Boku no Pico.... it was too twisted even for me.