Europe:
There are great views of Barcelona from Monjuic (particularly the cable car, or teleferic as it's called in Catalan) and from Tibidabo. I also liked the view from the MNAC mentioned above, and another nearby interesting view is from the roof of the Arena shopping mall nearby (the one that was put in an old bullring). There's an outside elevator at the Arena for extra cost, or you can ride the escalators inside to the roof for free.
The views of Florence from Fiesole are fantastic, and it's only a local bus ride to get there.
In Budapest, the Buda side has great views of the Pest side.
In Lucca, you can walk on the walls for free (or rent bikes cheaply) and get great views of the city.
If it's raining or about to rain in Edinburgh (and that's not a rare occurrence), the views of the castle from Prince's Street are very dramatic.
La Turbie, on the French Riviera, has amazing views of Monaco. If you don't have a car, buses only run there a few times a day, so you do have to check schedules carefully to make sure you're not stranded.
If you don't mind climbing something like 400 steps, the central tower in Bologna rewards you with a great view of the city's intact medieval street plan, with its "bicycle wheel and spokes" layout. It's not free but I believe it's inexpensive.
I second the Milan Cathedral - amazing!
Canada:
In Montreal, you can climb Mt Royal for free (but it's quite steep), or take a city bus there, then go to the Chalet Lookout. If you're taking the bus, it doesn't stop right at the chalet - be sure not to miss the stop.
In Vancouver, the SeaBus takes you to North Vancouver, with great views of downtown.
United States:
In Pittsburgh, you can take public transport or walk from downtown across the Smithfield Street Bridge to the Monongahela Incline, then take it up for a great view of downtown.
In New York City, the Brooklyn Heights Promenade, just across the Brooklyn Bridge, offers a great view of Manhattan (sadly, now mildly obstructed by some Brooklyn waterfront development, but still wonderful).
San Francisco has great views from all kinds of places - no need to pay a lot, just keep your eyes open as you go about town.
Roman Polanski is supposed to be the originator of the phrase, "Los Angeles is the most beautiful city in the world - provided you see it at night, and from a distance." This sounds like an insult, but it's not. It was decades ago, but I still remember flying into LA at night and how beautiful it looked from the airplane window - an undulating carpet of lights that went on forever. So, if you can wrangle a flight arrival or departure from Los Angeles at the correct time, you get a great view of the city for free. The more standard way to get these views is from Mulholland Drive.
Great topic!