Botanical gardens in or close to Paris, France

The other answer is very thorough but perhaps you would also like to look at Bois de Vincennes. It is located in Eastern Paris and has a zoo, a botanical garden, a Buddhist temple and a children’s farm along with many grassy fields and lakes.

It is very accessible from public transport and bikes are available for rent to explore its vast confines.

More info at Paris Tourism

There is also an exhibit of rare or forgotten species of apples, in the Jardin de Luxembourg. It is a rather humble park by Paris standards, but it is my favourite.

I would recommend Claude Monet's garden in Giverny. It is not too far away from Paris, and there are bus tours available. Giverny Monet's Garden

Rather than Versailles and its crowd, if you'd like to see some classic Le Notre garden I'd recommend Vaux le Vicomte, which also comes with a lovely and much more accessible castle https://www.gardenvisit.com/gardens/chateau_de_vaux-le-vicomte

Within Paris itself my favorite is the Parc Monceau, an English-style garden in the 8th arrondissement. Not so much.. gravel :-). But for a more formal botanical approach, Bagatelle in the 16th is a gem https://www.french-gardens.com/gardens/bagatelle.php

I like the musée départemental Albert-Kahn. It is located in Boulogne Billancourt, right outside Paris.

It has beautiful gardens: French, Japanese and a few "forests".

Warning: it is currently (September 2018) under renovation and the gardens are closed. I could not find an expected date for the end of the renovation, please feel free to remove this warning if it is over :)