na-adjective, noun
excessive; undue; unmerited; beyond one's station
Describes something that exceeds what is appropriate, deserved, or within one's proper limits. Often used with nouns like 望み (hope), 幸せ (happiness), or 待遇 (treatment). Can carry a nuance of 'more than one deserves' or 'presumptuous'.
彼は分外の望みを抱いている。
He harbors excessive ambitions.
これは私には分外の幸せです。
This is more happiness than I deserve.
Compound of 分 (ぶん, 'portion, share, one's lot') and 外 (がい, 'outside'). Literally 'outside one's share', giving the meaning of something beyond what is proper or deserved.