ssh inside ssh asks for "Enter passphrase for key..."
 

I have keys setup between two machines. When I ssh in to the first machine, it goes straight in without me having to type my password because I have my key as an authorized key.

Now, that machine that I ssh'ed in to also has its key authorized in the next machine I'm about to SSH in to, however it asks me:

Why do I need to enter the passphrase, and how can I avoid this?

It looks like when you created the rsa key on the second machine, you typed a passphrase for it instead of leaving it blank.