Holomorphic function on a unbounded domain

I have problem in understanging the following question

Q. Show by means of an example that a non-constant holomorphic function on a n unbounded domain need not achieve it's maximum modulus on the boundary of that domain

Answer: here we assume that tha boundary is not empty.

Maximum Modulus Principle says that

*"|f(z)| can only achieve its maximum value on the boundary unless it is constant"*

So i am wondering how can i connect the MMP with the above question.

Any help will be appreciated....Thanks