How to avoid iTunes asking for iCloud account? iTunes always asks me to log into my iCloud account. I do not want to use iCloud. This message to log in drives me crazy, it pops up every five minutes. How can I stop/avoid this, while still using iTunes?

To make it clear: I do not want to connect to Apple services all the time. I have an Apple ID, but when I created it, it was for the online service. I have bought music on iTunes, but after testing it I went to only buy DRM free music somewhere. However, I like the hardware, I like many things regarding Mac OS, but I have my own clouds and I do not want to connect to some internet service all the time.

There is a similar question here: iTunes asking for password for wrong iTunes account
iTunes is requesting that you sign in with your Apple ID. Your Apple ID is used to associate your iTunes purchases and credit details with iTunes. 

An iCloud account is not the same as an iTunes account, but typically, one uses the same Apple ID for both. However, it is possible for one to use a DIFFERENT Apple ID for iCloud and iTunes. In fact, this is quite normal for those that shared iTunes purchases with other family members before Apple implemented Family Sharing.

The answer is simple: create separate Apple IDs for iCloud or iTunes. If your Apple ID is currently used for iCloud, feel free to create a new one just for iTunes. 

Note that iTunes doesn't use iCloud for music, or storage, or really anything, so it is not clear what is the basis for your objection.