Alongside the ubiquitous complex period $2\pi i$, there's also the p-adic analogue $t$ which is a uniformiser in Fontaine's period ring $B_{dR}^+$ (see for instance anything written by Colmez e.g. this) as well as characteristic $p$ analogues such as the Carlitz period (see e.g. notes of Brownawell and Papanikolas).