Got is very informal. I wouldn't use either for present situation without the auxiliary have: I've understood (or I understand) or I've got it.
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Perfect StrangerAnd do people ever say: I've gotten it?Some do. It can mean "I have (already) obtained it (or received it, or bought it, etc.)". Some people regard it as an Americanism to use gotten as the past participle of get. Others dispute this, saying it's also possible in British English. It isn't used to mean "I understand", however.