Back in the day, it was as easy as going to a magazine display rack and tearing out all the subscription cards. I'm not sure if that still works today. Do they still send the magazines before they get your money? I haven't subscribed to a magazine or tried doing that to someone in years, so I'm not sure how it all works now. Since the magazine business is quickly dying, I doubt they're as generous as they used to be.
If you want to annoy someone with junk mail, start signing them up for as many things as you can. When you're passing through a store and see an amazing offer, sign them up. At the malls when they have a new car on display as a raffle you can enter, sign them up. I filled out a few of those car raffles legitimately and noticed a huge increase in junk mail and phone calls because of it. Sign them up for credit cards using the computer kiosks you see in all the stores. Sign them up for catalogs online. Church stuff, since those people will never leave you alone.
Personally I think all these methods are horribly inefficient since you will probably spend more time filling out things in their name than they'll spend throwing away junk mail and hanging up on telemarketers. But if you're really dedicated to being a mild nuisance to someone, then this will do it.