A colloquial emphatic negative suffix (attached to the -masu stem of verbs) meaning there is no way something will happen / absolutely cannot do ~; a strong spoken denial.
そんな難しい問題を君ひとりで解けっこないよ。
There's no way you can solve such a difficult problem by yourself.
明日までに全部終わらせっこないって、計画が甘すぎる。
There's no way you'll finish everything by tomorrow — that plan is too optimistic.
彼が謝りっこないから、放っておいたほうがいい。
He'll never apologize, so it's better to leave it be.