extremely busy with many matters
A yojijukugo often used humorously or self-deprecatingly to describe being swamped with various tasks or official duties.
Lately I've been so busy with various things that I can hardly take a break.
"You seem busy." "Yes, I'm swamped with work."
多忙 is a straightforward, neutral term for 'busy', while 御用繁多 is a more formal or playful yojijukugo emphasizing being occupied with many duties.
A yojijukugo combining 御用 (official business, matters) and 繁多 (numerous, busy). The exact historical derivation is uncertain, but it is used to express being overwhelmed with many tasks.