noun
leaving everything up to someone else; entrusting completely
Often implies a hands-off approach where one person delegates entirely to another, sometimes with a nuance of neglect or over-reliance.
彼に任せっきりで、私は何もしていない。
I've left everything up to him and haven't done anything myself.
仕事を部下に任せっきりにするのは良くない。
It's not good to leave all the work to your subordinates.
任せる is the base verb meaning 'to entrust'; 任せっきり is a noun/adverb form emphasizing the state of having left everything to someone.
From the verb 任せる (to entrust) + っきり (emphatic suffix indicating 'only' or 'completely').