Nero and 20 for the ask game! But no HIVE staff and no Diabolus, they’re too easy :D
I don't feel like doing the thing I am supposed to be doing (writing) so you are getting a long answer to a short question.
To me, the clause that tells us shared screentime is irrelevant suggests that the premise of this question suggests a query into compatibility above all. So, for example, though a character named Anastasia Furan may have canonically committed certain actions that would eliminate her from Nero's Friend Pool, those actions are not an inherent justification for discrimination.
The issue of compatibility then becomes an interesting one because one does not need to be exactly like one's friends in order to get along with them. Indeed, some level of difference is preferred to balance the act. This of course is evident in Nero's (canon) friendship with Diabolus Darkdoom, a man who is way more WooHoo than Nero, but in so doing is the balance to add more levity, action, and gadgetry to the narrative than Nero provides on his own.
However, there is also a line after which significant difference does result in incompatibility. This is why, spoiler, Anastasia Furan will not be winning the gauntlet. But she is in the pool anyway out of a misguided interest in thoroughness.
The pool I will draw from is as follows:
Mrs. McReedy, Mr. Littleton, Lao, Number One, Otto, Wing, Laura, Nigel, Franz, Shelby, Block, Tackle, Tom, Penny, Madam Mortis, Cypher, Xiu Mei, Overlord, Gregori Leonov, Sebastian Trent, Constance, Verity, Jason Drake, President Obama, Pietor Furan, Esteban, Cole Harrington, Duncan Cavendish, Anastasia Furan, Nathaniel, Joseph Wright, Lin Feng, Carlos Chavez, Zero, Queen of Shadows, Anna
I shall identify Key Nero Traits and use these as a basis of elimination. Whoever remains at the end of the list are the best friends*.
*Within the given criteria. Obviously the most correct answers are Raven, Darkdoom, and the Contessa, all of whom are disqualified for being HIVE staff or themselves.
Onward. Traits essential to Nero include:
intellect and culture (Don Giovanni, Tchaikovsky, dance, Dickens)
Characters eliminated: Block, Tackle, Mrs. McReedy
Nero is actually not a huge asshole about these sorts of things, really, but I do think we can all agree there is a line here that he won't cross.
more humble beginnings (hot wiring, Nathaniel's whole vibe, Sex Pistols)
Characters eliminated: Jason Drake
There are hints throughout the book that much of who we see Nero as is a carefully constructed facade. With that in mind I do think Nero wouldn't value a friend who has had an easy path lubricated by wealth and fame.
indulgent authoritarian (very strict and rigid structure at HIVE, with efforts to make it a fun and normal place, such as access to movies, team sports, recreation facilities, wrapped snacks, the ability to make popcorn on demand)
Characters eliminated: Otto, Wing, Laura, Shelby, Lucy, Tom, Penny
I personally think that between each character's personal values and fate, as well as the Bloodline epilogue, there will be a part of each of them that is grateful for their time at H.I.V.E. (or grateful for who they became as a result, perhaps) and another part that will never be able to forgive Nero for the experience either. Not necessarily in a dramatic way but probably in a real way.
However, this is not a deal breaker for two particular students. So they get to move forward.
deeply loyal and serious about his role as a caretaker (voicing an interest in his students' right to privacy, will risk his own safety if someone in his care is in danger, makes Glasshouse rescues twice)
Characters eliminated: Overlord, Esteban, Sebastian Trent, Pietor Furan, Cole Harrington, Duncan Cavendish, Anastasia Furan, Joseph Wright, Lin Feng, Carlos Chavez, Anna
Nero is so serious about giving people a good life, and the Glasshouse standards in particular as well as Wright's behavior in book 9, to say nothing of all the ways these people put his students' LIVES in danger just makes these super not viable.
Also, let's be real: anyone who doesn't implicitly respect Raven is never ever ever gonna be friends with Max Nero. Period.
man of evil principles (kidnaps children, imprisons people because he feels like it, keeps a personal murderer on staff, prefers the Sex Pistols over the Clash)
Characters eliminated: Mr. Littleton, President Obama
Dr. Nero can talk to normies but normies cannot talk to him.
powerful and negligent (given significant recognition and deference in council settings, but has multiple issues slip through the cracks--Overlord early start, never carries a weapon, kidnapping in book 3, tragedy in book 6, issues with security in book 7, etc.)
Eliminated characters: Lao, Cypher, Number One, Queen of Shadows
These three would have the most to say, but. Again. Different reasons. However, I think all three would be critical enough to make a friendship impossible.
prefers to be a big fish in a small pond but who knows other big fish (choosing HIVE over council responsibilities)
Characters eliminated: Madam Mortis, Constance, Verity, Zero
I think the vibe I get from the H.I.V.E. staff and people we see in the world is a Nero who loves meeting experts in other fields. This is a hot take but if he had met Raven as a teenager and she decided to leave behind her life of villainy and then become a high school administrator instead, and then was better at it than him??? I genuinely think he might have had a crisis about it. Anyways henchmen are unlikely to interest him.
Too Family Membery (personal opinion)
Characters Eliminated: Nathaniel, Nigel, Franz
Obviously I write fan fictions about this exact concept and I don't REALLY have textual stuff here other than the fact that I just know these three are vetoed. Nathaniel and Nero have a complex relationship with hurt and respect on both sides and they aren’t best friends but also they can't be because it's way deeper than that, too. And then Nigel and Franz??? In what world are these two clowns going to be a major emotional support to the big man? They're his baby employees like come on now.
And so after all that I am delighted to announce our winners:
Xiu Mei and Gregori Leonov!
I think both of these characters are villainous enough to be interesting friends while also respectful of Nero's primary vocation as a teacher and a guardian.
Is it cheating to grant the victory to two people he was arguably already friends with? Maybe. But as an evil teacher he has to approve of SOME cheating.