I want to experiment/play around with non-relational databases, it'd be best if the solution was:
What can you recommend for me?
Essentially, I would like to be able to install this system on a shared linux server where I have little user privileges.
I recommend you consider BerkelyDB with awareness of the licensing issues.
I am getting very tired of people recommending BerkleyDB without qualification - you can only distribute BDB systems under GPL or some unknown and not publicly visible licensing fee from Oracle.
For "local" playing around where it is not in use by external parties, it's probably a good idea. Just be aware that there is a license waiting to bite you.
This is also a reminder that it is a good idea when asking for technology recommendations to say whether or not GPL is acceptable.
From my own question about a portable C API database, whilst a range of other products were suggested, none of the embedded ones have Python bindings.