You can give your info to the registrar and then make it anonymous to whois domain.tld searches so it’s not public. Cloudflare is the registrar I use these days because it’s a one-stop shop and used the company address but, at least in the U.S., they need your info for both credit/debit card processing. (Processing fees are cheaper the more info they provide but usually any address with the same zip code is enough.)

If you have nefarious plans, I don’t have a good recommendation. But if it’s just about privacy, I don’t know if it’s really possible to be completely anonymous anyway. I guess you could use a gift card or something but at least in the U.S., if you own or buy a house, your address is public info already anyway. Shit, city hall will probably give you blueprints of any house.