<p>At what point does a MySQL database start to lose performance?</p>

<ul>
<li>Does physical database size matter?</li>
<li>Do number of records matter?</li>
<li>Is any performance degradation linear or exponential?</li>
</ul>

<p>I have what I believe to be a large database, with roughly 15M records which take up almost 2GB. Based on these numbers, is there any incentive for me to clean the data out, or am I safe to allow it to continue scaling for a few more years?</p>

The physical database size doesn't matter.  The number of records don't matter.
In my experience the biggest problem that you are going to run in to is not size, but the number of queries you can handle at a time.  Most likely you are going to have to move to a master/slave configuration so that the read queries can run against the slaves and the write queries run against the master.  However if you are not ready for this yet, you can always tweak your indexes for the queries you are running to speed up the response times.  Also there is a lot of tweaking you can do to the network stack and kernel in Linux that will help.
I have had mine get up to 10GB, with only a moderate number of connections and it handled the requests just fine.
I would focus first on your indexes, then have a server admin look at your OS, and if all that doesn't help it might be time to implement a master/slave configuration.