The Eiffel Tower is the most recognized structure in the world, and seeing it does not require a ticket. It was built for the 1889 World’s Fair by Gustave Eiffel’s company and was meant to be temporary. It stayed, kept around at first because it made a useful radio mast. Climbing or riding the elevator is paid and tiered: taking the stairs to the second floor is the cheapest option, the lift costs more, and going all the way to the summit costs the most. Skipping the ascent entirely is what most budget travelers do, and they rarely regret it. The reason is simple. Paying buys the view out from the tower, but the view of the tower, the one everyone actually wants in their photos, is free.
The best free vantage point is the Trocadero terrace across the Seine. It frames the whole structure head-on and costs nothing. The other one is the Champ de Mars, the long park stretching out behind the legs. Budget travelers sit on the Champ de Mars with a baguette and a bottle of wine from a supermarket and look up. That picnic costs almost nothing and is the thing most people remember long after the trip. The ground level under the four iron legs is also free to walk through, so getting close enough to crane your neck up the ironwork never costs a cent either.
The tower sits in the 7th arrondissement and is easy to fold into a free walking day along the river. From here it is a short walk along the Seine toward the bridges, and central Paris landmarks are all reachable on foot or one short metro hop: the Louvre Museum, the Arc de Triomphe at the top of the Champs-Elysees, Notre-Dame Cathedral and the riverside lanes of Ile de la Cite, and the cafe-lined streets of Le Marais. Several of them, the river path included, are free to enjoy from the outside, which means a full day of Paris sightseeing can come close to costing nothing.
A tip on timing: come at dusk and stay a few minutes after dark. On the hour, every hour, the tower breaks into a sparkling light show for about five minutes. It is one of the best free sights in Paris and you only need a patch of grass to watch it.