Somewhat adventurously, we went to Eiffel Tower and Arc de Triomphe the same evening, and both visits involved climbing a lot of stairs. A LOT of stairs.
You cross an event horizon bound by four legs of iron lace, and suddenly you no longer see the building that just a moment ago was the tallest thing visible: you are looking at it from below. But this eye of the singularity, the courtyard underneath the Eiffel Tower, is bubbling with mundane activity. There are food and beverage stands and knick-knack shops. The crowd is thick with vendors carrying chains with dozens of miniature Eiffel Towers strung on them. Those vendors are probably illegal, because when police arrives, they scatter like bugs.
Miniature models of Eiffel Tower are also sold at other famous tourist points, such as Arc de Triomphe and Sacre Coeur Basilica at Monmartre. But I never saw anyone sell models of those other objects -- maybe they are not considered as recognizeable.
Crossing into the Eiffel Tower courtyard you immediately see lines at the elevators. Those lines are long. Hours-long. However, observation decks in Eiffel Tower are accessible by stairs. The decks are at Level 1 and Level 2 of the base, which corresponds to 23rd and 46th floors of an ordinary building. The stairs don't go the top: for that, you'd have to take an elevator. Given the lines at the elevators, and our plans to visit Arc de Triomphe the same evening, we chose the stairs. Walking 46 floors to Level 2 was... something else. But we survived.
While all cities look pretty much the same from above, there are plenty of aids to identify the objects you see. Every 20-30 meters around the perimeter of the observation deck there are binoculars for your viewing pleasure, and a plaque that explains what are the most prominent objects in sight. Still, it inevitably turns out that some flashy palace that draws your attention is only of minor significance; whereas the most famous objects are so thoroughly lost in the chaos of geometric shapes that you have to look long and hard to find them.
Immediately after Eiffel Tower we went to Arc de Triomphe. We got there at twilight, and climbed 247 steps to the top. I don't know what floor of a typical building it compares to, but it's pretty high. We could not put it off for another day, because we already tried to visit Arc de Triomphe twice, and failed. The first time we underestimated just how long it takes to get from any place to any other place in Paris by Metro. Staying near the Southern edge of Paris, it took at least half an hour to get anywhere "interesting". Most points of interest were 2-3 Metro rides away, taking as much as an hour to get to.
The second time we arrived at Arc de Triomphe only to find it closed for the Charles de Gaulle day (June 18th) ceremony. The ceremony took place under the Arc itself, and involved a bunch of VIPs. Rumors said France's president Sarkozy was there. We stood too far from the Arc to make out any faces. The Arc is surrounded by a traffic circle, and is reachable only by underground passageways. The police had blocked off those passageways. They had not, however, redirected the traffic away from the traffic circle. That was odd. What's to keep a terrorist from a drive-by bombing?
A crowd had gathered on the outer side of the traffic circle to watch ant-sized dignitaries go through the ceremony (which involved speeches and not much else). A bunch of friendly 20-something French guys were passing around a flask with, I guess, hard liquor. We asked them what kind of holiday was Charles de Gaulle day, and they were eager to explain, but their English wasn't good enough for that. A convoluted, halting explanation involved World War II and England, and Charles de Gaulle doing something special. Later I found this article from The Guardian about the significance of that day; it also says Sarkozy was in the UK on June 18th of 2010, so he could not have been at the Arc de Triomphe.
More pictures from my trip to Paris are in my photo gallery.