I would like to access a user's Google Plus One history
With +1 enabled, the history is saved in your Google profile and optionally can be displayed:
http://www.google.com/+1/button/
It is possible to access this programatically (once the user has given permission via normal Google Authentication and Authorization?)
I have only been able to find information for the API to add the button to sites.
You can allways try to parse the data used by G+ itself.
The G+ user profile has a tab with all public +1, that can be fetched with
https://plus.google.com/_/plusone/get?oid=<google-plus-id>
It seems to be related to JSON, but with some differences.