The late Professor Jonathan Borwein was fascinated by the constant $\pi$. Some of his talks on this topic can be found on the CARMA website.
This homage to Jon is based on my talk at the Jonathan Borwein Commemorative Conference. I will describe some algorithms for the high-precision computation of $\pi$ and the elementary functions, with particular reference to the book Pi and the AGM by Jon and his brother Peter Borwein.
Here "AGM" is the arithmetic-geometric mean of Gauss and Legendre. Because the AGM has second-order convergence, it can be combined with FFT-based fast multiplication algorithms to give fast algorithms for the \hbox{$n$-bit} computation of $\pi$.
I will survey a few of the results and algorithms that were of interest to Jon. In several cases they were either discovered or improved by him. If time permits, I will also mention some new results that would have been of interest to Jon.