What can I do to increase the performance/speed of my PHP scripts without installing software on my servers?
Follow some of the other advice first like profiling and making good resource allocation decisions, e.g. caching.
Also, take into account the performance of outside resources like your database. In MySQL you can check the slow query log for example. In addition make sure you didn't design your database an forget about it. Optimizing your queries (again for MySQL) against real data can pay of big.