I've probably spent months of my life climbing to the top of the Eiffel Tower every day with groups of tourists. The problem? We would spend what felt like forever waiting to get up there, and I could feel their disappointment when they arrived.
They'd look out across the city, and the view wouldn't mean much to them because you can't see Paris' most iconic landmark if you're standing on top of it. Sometimes, you can't even see very far out if there's morning fog or rain.
Instead, I recommend climbing the Montparnasse Tower or going to the rooftop café at Printemps, a French department store, to look out across the city.
For a special treat, I always tell tourists to get to a viewpoint at night, when the Eiffel Tower sparkles for five minutes at the top of each hour.