Depends entirely on the person and what they like.

This is no different than asking someone for top three tv shows for someone new to TV.

But that being said.

Mushishi https://anilist.co/anime/457/MUSHISHI/

Supernatural, mystery vibe.

HUNTER×HUNTER (2011) https://anilist.co/anime/11061/HUNTERHUNTER-2011/

Cult classic, superpowers & adventure.

Yuru Yuri https://anilist.co/anime/10495/Yuru-Yuri/

Slice of life, school-life, cute

Anime is a medium rather than a genre. The "ultimate anime recommendation flowchart"s that pop up now and then are a pretty good place to start with different tastes in mind.

It really depends on what they are into.
Ideally it will be something short and self contained. I’ll start with the gold standard for recommendations: “Cowboy Bebop”

If they want something light-hearted and comedic, “Spy X Family” is a good one
“Vinland Saga” for something heavier and more serious

For fantasy fans there’s “Frieren” and “Delicious in Dungeon”, even if both are part 1 of a larger story
if they want fantasy aimed at someone younger, then “Little Witch Academia” is a good pick

If they want a combo of sports and romance, I thought “Yuri on Ice” was top tier.

With those out of the way, since you only said 3, I’ll pick these 3 to give a wide variety of what anime has to offer:
Cowboy Bebop, Delicious in Dungeon, Spy X Family

The anime spectrum is so vast that the safest bet is recommending a Miyazaki film like Spirited Away or Ponyo. Then, just when they’re hooked, you hit ‘em with Grave of the Fireflies.

This is a tough one. If I was recommending anime to someone brand new to anime then there are a few guidelines I’d follow:

• Can’t take too long to develop. So while things like Frieren are undoubtedly amazing, the slow start means I wouldn’t recommend it to a Newbie.
• Must be palitable for a wide range of audiences. Panty and Stocking, Elfen Lied, and Devilman Crybaby might be a great shows but they’re pretty in-your-face and could be off-putting to people.
• Must be easily digestible. One Piece might be a great anime but no one is going to want to start on a series with over a thousand episodes. Likewise stuff like Serial Experiments Lain and Evangelion are good but far too out there for a first timer.

To that end I usually start people with Your Name. It’s movie length so easily digestible. The animation is beautiful. The plot is relatively unique while still being easy to understand. And pretty much anyone can enjoy the mix of slice-of-life and supernatural.

I’ve also had a lot of success with Ouran High School Host Club. Everyone can understand the concept of a school but then the inclusion of the wacky rich people makes it a fun time. Another school slice-of-life could work here too like Toradora or Azumanga Daioh.

Finally, back in the day I got a lot of people into anime with Beck. It’s still pretty good even by today’s standards but another anime about music or a band would also work since people tend to understand music or band dynamics in general. Possibilities include K-On or Your Lie in April.

Anyway, just my thoughts.

Can’t not list Fullmetal Alchemist: Brotherhood

For a gaming anime, either Shangri-La Frontier or Log Horizon

For romance, either HoriMiya or The Angel Nextdoor Spoils me Rotten

For god-tier binge watchers that are okay waiting for a long term payoff of storytelling: One Piece.

If I were to introduce someone into anime, I would introduce them a classic anime. Something like the Dragonball franchise.

Then I would show them a what I call a modern classic. Something like Naruto.

Lastly an anime that is actually new from scratch that looks like it might be a new long lasting anime series.

So you hit some of the classic animes and modern animes.

Akira

Ghost in the Shell

One Punch Man

I would suggest you start with Studio Ghibli/ Mr. Miyazaki’s works

• My Neighbor Totoro
• Princess Mononoke
• Spirited Away
• Pom Poko
• Castle in the Sky
• Only Yesterday

You may also want to watch Makoto Shinkai’s works

• 5 Centimeters per Second
• Your Name
• Weathering with You
• Suzume

If someone has never really seen any anime before, I think Mushishi is a good one to recommend to people. It doesn’t contain any of the less-palatable anime tropes, can be enjoyed by any age group, and is just overall really chill and interesting. It’s also episodic, so if they watch 2 or 3 episodes and decide it’s not for them, they still get a full story without missing out on anything.

Echoing others’ thoughts that it very much depends on what the person enjoys in general. But it’s very hard to go wrong with Cowboy Bebop. If they lean towards fantasy, Record of Lodoss War. If they want a movie instead of a series, something from Miyazaki/Ghibli.

Depending on what they like, I’d probably tailor my recommendations i.e. if they’re more into action, drama, comedy, sci-fi, slice of life, etc.

In general though, my go to’s are:

• Paprika
• Cowboy Bebop
• Steins;Gate

Difficult, just spit balling recs:

Gurren Lagann

Cowboy Bebop

Your Lie in April

Bonus for the cultured: Keijo!!! it has earned each of those exclamations.

• Solo Leveling
• Apothecary Diaries
• Demon Slayer

It really depends on the person. You honestly couldn’t go wrong with giving them the MAL top 25 all time and having them read the description to see what fits their tastes.

I think some must watch shows, not necessarily in the top all time:

Erased

Psycho Pass Season 1

Akudama Drive