I did’t know much about the German keyboard layout but I know the Czech one, which is derived from it (we both use QWERTZ) and was able to look up most of what I didn’t know.
So, the keyboard has 4 layers: default, Shift, AltGr, AltGr+Shift (the fourth one is not standard but is recognized by xkb; in Czech I use it for custom character mappings, in German it is standardized but Linux-only).
Default layer prints lowercase letters a-z and äöüß, numbers and the symbols in the lower-left of each key.
Shift layer prints uppercase letters A-Z and ÄÖÜ and symbols at the top left of each key.
Caps Lock only affects letters.
AltGr layer prints lower-right symbols, most of which are only populated in a later version of the layout.
As you can see, AltGr+2 produces ², and AltGr+3 produces ³. I think the full-size “2” and “n” are misprints. My old Czech keyboard has some errors too.
we have lots more diacritics so the number row only prints numbers on its Shift layer (most people therefore use the numpad only)
to print rare diacritics (ó, ď, ť, ň, and German ä, ö, ü), one has to first press the corresponding modifier key (´, ˇ, ˚, ¨) like on typewriters
an alternative for common capital diacritics (á, é, ě, í, ú, ů, ý, ž, š, č, ř) is to briefly turn on Caps Lock (advantage over typewriters)
pressing the ˚ key twice prints the degree sign (°) twice (Windows) or once (Linux)
there is a bloody dedicated § key but we need to press AltGr+7 twice, then backspace (or Alt+96) for a grave (`), which is part of ASCII and used in Markdown
physical keyboards almost always reserve the right side of the keys for the English-US layout (very confusing for novices) so one has to type in the AltGr layer blind (except for €); it contains useful symbols ([]{}<>|\€$@#^&×÷`) as well as useless ones (Đđ – these are Slovene, why not the Slovak Ôô?), leading people to prefer Windows-only left-Alt+numpad codes (such as Alt+64 for @) that use the obsolete OEM-1252 codepage (the Unicode extension has to be enabled via registry and Alt+letters hex codes get passed to programs anyway, often defocusing the input element). I only found a Slovak one on Wikimedia Commons
some lazy manufacturers combine the Czech/English and Slovak/English layouts, which are similar except ľ, ť and ô, leading to 5 (!) symbols per key, 3 of which are irrelevant unless you switch layouts
Gboard for Android offers QWERTY for Czech, which looks normal (hold for diacritics, potentially swipe for ě and ů) and the unpopular QWERTZ-PC, which has all the physical keyboard’s quirks, but its “Czech QWERTZ” is based off German QWERTZ, containing ú and ů but not the other diacritics for some reason. All other keyboard apps with Czech language layout get this right (hold for diacritics, potentially swipe for ě and ů)!
I did’t know much about the German keyboard layout but I know the Czech one, which is derived from it (we both use QWERTZ) and was able to look up most of what I didn’t know.
So, the keyboard has 4 layers: default, Shift, AltGr, AltGr+Shift (the fourth one is not standard but is recognized by xkb; in Czech I use it for custom character mappings, in German it is standardized but Linux-only).
As you can see, AltGr+2 produces ², and AltGr+3 produces ³. I think the full-size “2” and “n” are misprints. My old Czech keyboard has some errors too.
By the way, Czech is more chaotic:
´,ˇ,˚,¨) like on typewriters˚key twice prints the degree sign (°) twice (Windows) or once (Linux)§key but we need to press AltGr+7 twice, then backspace (or Alt+96) for a grave (`), which is part of ASCII and used in Markdown€); it contains useful symbols ([]{}<>|\€$@#^&×÷`) as well as useless ones (Đđ – these are Slovene, why not the Slovak Ôô?), leading people to prefer Windows-only left-Alt+numpad codes (such as Alt+64 for @) that use the obsolete OEM-1252 codepage (the Unicode extension has to be enabled via registry and Alt+letters hex codes get passed to programs anyway, often defocusing the input element). I only found a Slovak one on Wikimedia Commons