How big can a MySQL database get before performance starts to degrade
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.
The database size does matter. If you have more than one table with more than a million records, then performance starts indeed to degrade. The number of records does of course affect the performance: MySQL can be slow with large tables. If you hit one million records you will get performance problems if the indices are not set right (for example no indices for fields in "WHERE statements" or "ON conditions" in joins). If you hit 10 million records, you will start to get performance problems even if you have all your indices right. Hardware upgrades - adding more memory and more processor power, especially memory - often help to reduce the most severe problems by increasing the performance again, at least to a certain degree. For example 37 signals went from 32 GB RAM to 128GB of RAM for the Basecamp database server.
In general this is a very subtle issue and not trivial whatsoever. I encourage you to read mysqlperformanceblog.com and High Performance MySQL. I really think there is no general answer for this.
I'm working on a project which has a MySQL database with almost 1TB of data. The most important scalability factor is RAM. If the indexes of your tables fit into memory and your queries are highly optimized, you can serve a reasonable amount of requests with a average machine.
The number of records do matter, depending of how your tables look like. It's a difference to have a lot of varchar fields or only a couple of ints or longs.
The physical size of the database matters as well: think of backups, for instance. Depending on your engine, your physical db files on grow, but don't shrink, for instance with innodb. So deleting a lot of rows, doesn't help to shrink your physical files.
There's a lot to this issues and as in a lot of cases the devil is in the details.
I'm currently managing a MySQL database on Amazon's cloud infrastructure that has grown to 160 GB. Query performance is fine. What has become a nightmare is backups, restores, adding slaves, or anything else that deals with the whole dataset, or even DDL on large tables. Getting a clean import of a dump file has become problematic. In order to make the process stable enough to automate, various choices needed to be made to prioritize stability over performance. If we ever had to recover from a disaster using a SQL backup, we'd be down for days.
Horizontally scaling SQL is also pretty painful, and in most cases leads to using it in ways you probably did not intend when you chose to put your data in SQL in the first place. Shards, read slaves, multi-master, et al, they are all really shitty solutions that add complexity to everything you ever do with the DB, and not one of them solves the problem; only mitigates it in some ways. I would strongly suggest looking at moving some of your data out of MySQL (or really any SQL) when you start approaching a dataset of a size where these types of things become an issue.
Update: a few years later, and our dataset has grown to about 800 GiB. In addition, we have a single table which is 200+ GiB and a few others in the 50-100 GiB range. Everything I said before holds. It still performs just fine, but the problems of running full dataset operations have become worse.