Moreso than most people, President Obama has a very good reason for personally disliking Donald Trump and the notion of him becoming President: Trump was the de facto leader of the "birther movement," during which he antagonized Obama for years by lying about having proof that Obama had faked his birth certificate. He spent years trying to de-legitimize Obama's presidency with a ridiculous conspiracy theory based on nothing - so, with his last day in office as President, Obama should totally do something to troll him (that wouldn't have crazy legal ramifications):
1. Resign and Make Joe Biden the President
This has been a popular suggestion across the 'net: make Vice President Joe Biden the ACTUAL President for what would be Obama's last day in office. Why? Well, for one, Joe Biden and Barack Obama are BFFs - more than perhaps any other President/VP in recent memory. Joe Biden speaks highly of Barack, often saying that he's his brother, and even gave him this adorable friendship bracelet:
Of course, you may have heard that Obama returned the favor (and then some) by bestowing the Presidential Medal of Freedom (with distinction) on his pal/veep recently. So, the two like each other a lot, and it would be a very friendly gesture (probably the nicest gesture POSSIBLE IN THE ENTIRE WORLD).
But if he REALLY wants to go for it and cement himself as a true BFF, he'll step down from the presidency a day early and invoke the 25th Amendment to the constitution, which would make Joe Biden the 45th President of the United States (for only a day, but still). And while this would mostly be a really nice thing for Joe Biden (who had presidential aspirations of his own before agreeing to be Obama's VP), it would mainly serve as a great troll on Trump. Donald Trump already has a ton of "45th President" merchandise for himself - but if Joe Biden were to become the 45th, that would make Trump the 46th President, and all of his merchandise would be worthless.
And if there's one thing Trump hates, it's people showing disrespect towards his merchandise:
2. Claim Squatters Rights
Sure, Obama can't continue on as President after January 20th, but that doesn't mean he has to leave the White House. In Washington DC, you can claim "squatter's rights" and stay on a piece of property for six months before anyone can legally evict you. In the meantime, Obama would likely be restricted from accessing most of the White House, but could definitely annoy President Trump by hanging out in his pajamas and tweeting pics of himself giving noogies to Trump staffers.
It would be the funniest case of squatters' rights outside of Dennis Duffy:
3. Get (or Give, the Choice Is Up To Michelle) a Golden Shower In the Lincoln Bedroom
You may have heard the unverified rumors that Donald Trump received a golden shower from prostitutes in Moscow, specifying that the sexual act take place in the same room that President Obama and the First Lady had stayed in recently. Of course, these rumors may be completely false and untrue, but one thing is not changed by them: they pissed Donald Trump off (pardon the pun). He doesn't like people mocking him in such a widespread manner, so this story REALLY got under his skin - the point where discussion of it largely dominated his first press conference as President-Elect.
What is also not changed is that Barack Obama has good reason to dislike Donald Trump (again, the whole "birther" thing, to say nothing of Trump's political ambitions to under much of Obama's achievements as President). So a fun little thing he and Michelle could do in their final night in the Lincoln Bedroom is to have some piss-filled golden-shower-style intercourse...in a bed they know Trump will be sleeping in soon.
Of course, the bed would be washed and cleaned by the White House's cleaning crew (Obama should tip them pretty well if he does this), but Trump will have to sleep in it knowing Obama pissed/got pissed on in that bed. And even if the story about Trump is false, Trump is a notorious germophobe, and being unable to sleep soundly in the Lincoln Bedroom without squirming at the thought of the piss molecules that could be getting on him would make it all worth it.
4. Invite Actual Bruce Springsteen To Play at The White House
Trump's Inauguration talent list is looking pretty dire: the biggest act that's been booked is 3 Doors Down, a band that hasn't been relevant since the early 2000s. But things got REALLY embarrassing when the Trump Administration couldn't even manage to book a Bruce Springsteen COVER BAND (called "The B Street Band", a play off of Springsteen's E Street Band).
Luckily for Obama, ACTUAL Bruce Springsteen is a big fan of his, and has played for him frequently. So why not really rub salt in the wound of self-proclaimed dealmaker Trump by showing him up and having Bruce Springsteen play at the White House THE DAY BEFORE TRUMP'S INAUGURATION? That might sting a little.
The guy who ran for president on his ability to make and close deals could not successfully book a Bruce Springsteen cover band.-- Chase Mitchell (@ChaseMit) January 17, 2017
5. Say He's From Kenya, Just 'Cuz
Just to piss off Trump to the ultimate degree, steal every news headline and conversation by announcing he's a secret Muslim from Kenya the day before Trump's inauguration. Trump's already had to disavow his role in the birther movement, so he wouldn't be able to claim "Hey! I was right the whole time!" AND no one would pay attention to any of the press around his first day in office.
All in all, it'd be a pretty fun way to go out.