When you pay through your nose for something, you pay a heavy price for it. In other words, you pay too much for it. This informal expression has the same meaning as ‘cost an arm and a leg’.
*Never go to that restaurant. You have to pay through your nose for most things.
According to scholars, this idiom has been around for over ten centuries. I understand that when the Danes ruled Ireland, they imposed heavy taxes on the locals. If an Irishman failed to pay the tax, he had his nose cut off. In other words, if he didn’t pay, he had to literally pay through his nose.