What are the requirements of a nonce? Sometimes I read that a nonce has to be a random number but I disagree. A nonce just can't repeat itself. You could increase in by 1 every time if you are sure it would never repeat.
Yes, a nonce is a number not used more than once. In its purest sense there should be no other requirements than this, i.e. randomness or unpredictability should not be necessary. 

However, in certain settings stronger requirements are put on the nonces; like for instance in the CBC-mode of operation for block ciphers the IV (nonce) needs to be unpredictable (a requirement, when not followed, actually have led to real-life problems in the SSL/TLS-protocol:  Why is CBC with predictable IV considered insecure against chosen-plaintext attack?). 

See also: What is the main difference between a key, an IV and a nonce? for a more comprehensive discussion of the properties of nonces and IV's.