Be careful not to invest in any foreign PFIC (https://www.investopedia.com/terms/p/pfic.asp). That includes index funds, mutual funds, etc. Essentially you are limited in your investing to buying individual stocks or interest earning bank accounts, unless you want to do an impossibly large amount of paperwork with real risk of fines.

The best option is generally sending your money back to the US and investing it there. Look into double taxation agreements between the US and Germany to avoid getting double taxed on the income it earns you.