What can I do to increase the performance/speed of my PHP scripts without installing software on my servers?
1) Use latest version of PHP
The core team is working hard on improving the performance of PHP in every version.
2) Use a bytecode cache
Since PHP 5.5 a bytecode cache has been added to PHP named OPcache. Using OPcache can make a huge difference especially since PHP 7. It receives improvements in every PHP version and might even get a JIT implementation in the future.
3) Profiling
While developing profiling gives you great insight what exactly is happening. This helps a lot finding bottlenecks in your code.
One of the most used tools is XHProf but is not officially supported anymore and has issues with PHP >= 7. An alternative when you want to profile PHP >= 7 is Tideways which is a fork of XHProf.