La Basilique du Sacré Coeur
Featured in movies like Amélie and (anachronistically) Moulin Rouge, la Basilique du Sacré Coeur (Basilica of the Sacred Heart) is the highest point in Paris and one of the most easily recognizable landmarks from many points in the city. The basilica itself is beautiful, but I recommend coming up just for the view of Paris, which is one of the best you'll find anywhere. It's really worth seeing in daytime and/or at night.
While you're up there, check out the green bronze statue of Jeanne d'Arc — Joan of Arc — on the front right side of the church. The other guy over on the left is King Saint Louis IX. Also, around back and up on the roof, there's a great bronze statue of Saint Michel slaying a cute little dragon.
A cool nighttime hangout
At night, the steps up at the top of Sacré Coeur become a hangout for tons of local and foreign young people; crowds of kids sit on the stairs and drink, play guitars, sing, chat, and flirt. It's a very friendly, mellow scene that continues way into the wee hours, but I'd stay away if you're not a fan of Bob Marley, which seems to be the only thing any European kid with an acoustic guitar knows how to play.
Take the stairs or the funicular
Beware: the hike up to the church involves a few hundred stairs, but you can also take a funicular most (but not all) of the way, if you have a hard time with stairs or just if you're fabulously lazy. The funicular costs one Métro ticket.
Watch out for scam artists
Also beware: day or night, before you even get to the steps, the large open area down in front of the church, where the carousel is, is always full of a bunch of scam artists who try to force you to buy crappy bracelets. The scam is this: a guy blocks your path, with a big smile on his face, and tries to take your hand/arm. He distracts you with a little fast-talk (in whatever language you speak) about how these friendship bracelets are a tradition in his native country. Before you realize what's happening, the bracelet has been tied permanently onto your wrist and the guy is demanding ten Euros or something ridiculous like that. Don't let this happen to you! Don't stop to be polite, don't say no thank you, just keep walking, and if the guy blocks your path, just barge past him; he'll give up and find an easier target. I lived right next to Sacré Coeur for years, and while I never got taken by these guys, they sure as hell tried all the time. And unfortunately I saw tourists falling into this trap over and over and over; I once took a photo of the front of Sacré Coeur from the street down below and later noticed in the foreground there were FOUR tourists getting bracelets tied onto them by four of these bracelet jerks, all at the same time. Horrible. I don't know why the cops put up with it.
Take the scenic route
One way to avoid the stairs, and the bracelet guys too, is to take a long, winding stroll up and around the hill, so the elevation is much more gradual. Here's a Google map I made of this walk. Google says it takes about 25 minutes, but I'd say give it 45 and take your time.
Start on the street directly in front of Sacré Coeur, called Place Saint Pierre. Walk to your right and keep going straight 'til you arrive at the Place des Abbesses, with a Métro stop and a carousel (not to be confused with the carousel that's right in front of Sacré Coeur). Keep going straight and you'll join up with the Rue des Abbesses. Soon this street will fork, and you'll take the right side, called Rue Lepic. This street will curve to the right, and keep an eye out for a small plaque on your right side marking Vincent Van Gogh's old apartment! It's at 54 Rue Lepic. (Blog post with pictures here). Keep following Rue Lepic as it continues to climb up the hill and curve to the right. Eventually Rue Lepic will fork in front of a tiny triangular park, and you want to stay to the left on Place Jean-Baptiste Clément, still climbing upward on the hill. This part gets steep, but it's short. You'll hit a T-shaped intersection at the top of this hill, and you'll turn right on Rue Norvins, toward a bustling area known as the Place du Tertre. You're almost there! As you turn toward the Place, you'll see the top of Sacré Coeur up ahead behind some buildings. Make your way through the Place and you'll arrive behind Sacré Coeur on its Western side. Congrats! You're at the top of Paris!
Ooh, another thing: once you've cut through the Place de Tertre on your way toward the front of Sacré Coeur, you'll pass by another less-remarkable church tucked away in a small courtyard, called Saint-Pierre de Montmartre, which is absolutely worth visiting to see the 7th-century Roman columns inside. Check out my article about that here.
Pickpocketing and other crime
Notre Dame
An unfinished statue of the Virgin Mary