i don't want to do anything with a database, because... that'd be a lot of fields.
How many fields? Approximate idea? 10? 1000? 1,000 fields?
Without cookies you may sorta identify a mchine by a combination of IP Address+Operating System+Browser. That still is not 100% reliable for some pretty obvious reasons.
You may store some info on the server and regardless of the "number of fields" this is preferred to flat text files because of speed. The number of fields you are worried about is not going to change because you are trying to avoid a database. After all, if you are going to record some type of info about the user, the amount of that info is logically ordered into some kind of record (or array of strings, integers etc) i.e. equivalents of fields. You have to store information somewhere on the server. So you're going to have the same number of fields regardless what technology you use: cookies, flat text file or database.