What is the maximum version of iOS compatible with the 2nd-generation iPod Touch?

iOS 4.2.1 (8C148) is the maximum OS version it can be upgraded to so you won't be able to install 4.3 or newer OS using iTunes and Apple supplied versions of the software.

iOS 4.2.1 is the highest iTunes can install on that model. You can't get the latest greatest apps but there are tons of vintage apps available, even for models that can only run iOS 4.2.1 or earlier. The trick is finding them. This is one way, https://sites.google.com/site/appleclubfhs/support/advice-and-articles/finding-apps-for-older-devices Then there is an app to find these vintage apps, https://itunes.apple.com/us/app/vintapps-3.1.3/id639274910?mt=8&uo=4