Have you been before?
If not, strolling along the Seine, pavement cafes and Montmartre would be my top "non pay for entry" sights to see.
Sacre Coeur church has an amazing view over the city, or if you want gothic atmosphere, Sainte Chapelle has the most gob-smacking medieval stained glass. (Assume Notre Dame must still be closed after the fire?).
Unless you're into art, I wouldn't bother with the big galleries. If you are, you could easily spend all day there. However, I loved Monet's waterlillies at the Orangerie (small art gallery, big paintings) and there is sometimes something interesting to see at the Pompidou Centre.
French shopping - clothes, beauty etc - even at quite an ordinary level, is "tres chic" - however it's expensive at the best of times and will feel even more so after the Tory mini budget debacle of a couple of weeks ago.
There are some nice shops and atmosphere around the Marais district.
Eiffel Tower and Arc de Triomphe are iconic but pretty dumpy really. Enormous (hours long) queues for the catacombs will probably rule that out too, although it's interesting if you do wait.
My main takeaway from my last trip to Paris was just how expensive everything was. Budget for everything to be 1.5 x English prices, and perhaps even more now.