How many Chateaux is too many?

Well, Rick rates them 3, 2, and 1-star, on a scale of 0-3, in the same order that you've listed them. But I like to play these decisions by ear, after you have the Loire visit under your belt. If the two of you don't share the same love of chateaux, consider splitting up for one day's sightseeing. I'm not worried about you running out of things to do in Paris over 9 nights, unless there's also a question of how many museums are too many.

I love visiting chateaux and spend a lot of time at each one, so for me, 1-2 is ideal in a day. I think Vaux-le-Vicomte is amazing and absolutely worth a visit. It's one of my favorites.

I really enjoyed Fontainebleau because of it's history, but it's not a "wow" chateau imo.

Chantilly did nothing for me.

I'll admit that after Versailles and 4 Loire Valley chateaux (inside visits), I was a bit chateaud out and decided that I didn't need to go in any more of them. However, I was happy to go to a couple more just to photograph the exteriors, grounds, gardens, fountains, etc. I guess I would say, you'll know when you've had enough and I agree with Laura, if one of you is burned out on them and the other is not, it makes sense to split up for a day and each go their own way.

Nancy, is it "chateau'd" or "shattered" ??

If you have done Versailles, I would definitely look at Fontainebleau.
I would do Vaux-le-Vicomte ahead of Chantilly.
Chateau Vincennes, at the end of #1 metro line, makes a great (shortish) visit, especially if you hold a PMP.

Ten years ago I can remember visiting 4 châteaux during a week stay in the Loire region. After Chenonceaux, Chambord and Amboise I felt with Azay-le-Rideau it became a bit too much. I call it that I started suffering from chateau fatigue.
So I´m with the other posters, if you still want to add another during your stay in Paris I think Vaux-le-Vicomte is to prefer (in case you have seen enough interiors) as you can just visit the garden and enjoy the splendor of the whole setting. You can also think about Malmaison as the style is different so having a bit more variety.

I went to 3 châteaux in 8 days in Burgundy and Alsace. They were very different from each other; so each experience was new. All were beautiful with interesting histories. I got a brochure that described châteaux and there were others that I would have liked to see but couldn't fit in . . . next time. In your place, I'd do a little research into each one that you're considering. Then after the Loire Valley, decide if they are different enough to warrant your time.

These are all great suggestions. Thank you so much.

I have seen enough chateaux to last a lifetime, but I do think it is worth making the effort to see Vaux le Vicomte because of the gardens designed by Le Notre.
https://janettravels.wordpress.com/2010/09/20/vaux-le-vicomte-our-last-chateau/
We spent a whole day here. Be sure if you go to pay the extra couple of Euro for access to the Coupole atop the chateau; it is worth it.