This is a top-20 list for visiting Paris on a budget:
1) Walk around the Sacre Coeur basilica and the fun little shopping streets of Montmartre - free
2) Walk by the Moulin Rouge when it's dark enough to see the lights, but early enough that the scary people aren't out - free
3) Walk through the Jardin du Luxembourg to the street where the Pantheon is - free (although it's perhaps 4 euros to go inside the Pantheon)
4) Eat a gelato at Amorino - around 4 euros
5) Walk around and go inside the Notre Dame - free; climb up the belltower - a few euros (but that's such a cool Parisian thing to do)
6) Eat a crepe - prices range from 7 to 15 euros; to some extent you get what you pay for
7) View the Eiffel Tower from Place de Trocadero, particularly at night - free; then, get a macaron at my favorite macaron place, Carette - 2 euros; add the most amazing hot cocoa to the order - an additional 8 euros (but it's like drinking liquid chocolate)
8) Eat a pain de raisin or pain du chocolate for breakfast - 1,60 euros
9) Go to Le Pain Quotidien for a breakfast of bread and different chocolate and nut and fruit spreads - 7 euros, more if you add cheese, meat, hot cocoa
10) Stand under the Arc de Triomphe or the Eiffel Tower - free (it's probably worth it to pay to go to the top of the Eiffel Tower once; the view from the Arc is kind of cool over the Champs D'Elysees, but I'd probably save my money for something else)
11) Take a boat tour of the Seine - 11 euros, but only go with a company that gives each person their own headset
12) Go to the Pierre Lachaise Cemetary and look for the graves of famous dead people, or go to any other cemetary just for kicks because they are kind of cool
13) Ride the ferris wheel at the Place du Concorde - either 5 or 10 euros
14) Go to the following museums, in this order: Orangerie, d'Orsay, Louvre, Marmottan, Jacquemart-Andre - you can get a discount double whammy price for both the Orangerie and d'Orsay; if you want to have a couple days of just going to museums, the Paris museum pass gets you into a whole lot of places for one set rate, but you'd have to go from museum to museum to museum to make it worth it
15) Visit the gardens of the Rodin Museum - 1 or 2 euros (the museum itself is also quite nice; I think it's an additional 4 euros, but you should like sculpture because that's all they have there)
16) Hang out in front of the Centre George Pompidou, check out the fountains, and people watch - free
17) Read in the Tuilerie Gardens - free
18) Walk along the Seine and cross all of the bridges, my favorite (Pont Alexandre IV) - free
19) Check out the mini statue of liberty - free
20) Walk along Rue Sainte Honore to see where the super rich people shop (seriously, onesies for babies for over 100 euros) - free
Have a great trip and feel free to add to the list if you know of something or somewhere else to visit in Paris if on a budget.