If you go to their downloads page, you will find the following statement (my emphasis):
We recommend using the latest version of Python 3 [...]
I've been using Python 3.4.3+ in a Django 1.8.2 project without issues and I don't see why you should run into problems here.
However, if you do run into issues, you should file a bug.
What version is the most stable one today?
Whatever version shows up as the recommended download in the python homepage should be considered stable. If that happens to be v3.5, then you should use that.
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