I really like prototype.js and jQuery

It's possible to combine them because jQuery is well-written and it's easy to change $ to $j.

More libraries I like to use :

Scriptaculous and LivePipe user interface elements.

date.js is great for date parsing, formatting, and calculations

Flot for line and bar charts and FlotPie for pie charts (I hope one day they will merge!)

I made an improvement to jquery.flot.pie.js to extend the legend to be multiple columns so it can be used as a complete data table next to the chart.  See JavascriptFlot