Absolutely. You have to understand some things about Facebook architecture.
1. They hide cookies in their images.
2. They store data on your browser
3. They store data on your hard-drive
4. They monitor your device's audio
5. They can monitor your device's video
Facebook tracks you on two fronts: your physical device (browser and hard drive), and your internet location.
They use all of this data to track you for, you know, "relevant advertising." They actually have a patent out for this technology which they developed, I believe, with Google.
So in order to be completely anonymous, you'll actually have to take some considerable effort to evade their tracking. Are you ready? Here are the steps:
1. Buy a cheap, used computer from Craigslist and pay cash. Facebook has a payment processor and has the ability to track credit card payments or at least view them. To what degree I don't know, but take a precaution by paying cash.
2. Download a free VPN service (free also means your credit card and information wasn't used and this will enhance your anonymity). I like to use CyberGhost. It is not wise to use online proxy services because Facebook probably blocks most of them. But a VPN like CyberGhost runs on your desktop (or phone) and is invisible. This does two things: it changes your IP address, and encrypts your data. Not even your ISP can see what you're doing here.
3. Download Tor. Tor is another browser built on top of Firefox. The onion network was developed by the government to do anonymous internet communications. It has been opened up to the public. This will further encrypt and hide your location. Some Tor nodes are actually FBI or CIA-sponsored nodes, but they can only see a small portion of Tor traffic. Also, while you're running your other VPN, this will make it more difficult, even for government operatives. As a general rule, you should plan to hide yourself from government because that will ensure you're hiding yourself well from Facebook, who shares data with the NSA as we learned from Snowden and Zuckerburg's own admition.
4. You will have to partition your hard drive (at the very least) to run a Linux operating system. If you are new to Linux, I recommend Mint or Ubuntu because they will be more intuitive for you to use. Linux was built with privacy in mind. Mint is better-suited for privacy. If you're really paranoid, you'll probably do best with Whonix or Tails. Whonix runs every single thing through Tor by default. Tails can be run from an image on a CD, which means you don't need a hard drive. If you don't have a hard drive then facebook cannot store anything on it.
Note, you can also run it from a USB, but remember that USB's are storage devices. The reason why you must use Linux is because Billy Gates paid >$1 Billion to own ~2% of Facebook. Does he have access to some of Facebook's data? Is it shared with Microsoft? Ask these questions. If you do not know, assume they share information. But Linux is owned by no-one. Here is some proof from their own source code that they store data on your hard-drive:
As you can see, they use bootloader.js, which is a JavaScript library specifically built for storing data on local storages of your device, particularly when traditional cookies are limited and don't store enough.
5. Disable JavaScript. If you decide to use Tor, this will be automatically done for you. Otherwise, you can use Firefox (in private mode) and download the no-script add-on. Some functionality will be removed and much of Facebook may not work if you use it.
6. If you're using a notebook, get a business card and fold it in half. Place it over your device's camera. You can usually remove the screen of your laptop and unplug the camera. Alternatively, you can disable in in most modern operating systems, or do a combination of the three.
7. Buy a burner. Facebook will eventually attempt to "verify" you with a cell phone number. You will need to undermine this by buying a burner phone. You can go to Walmart and pay $10 for a Net-10 cell phone or Trackfone. They come with enough minutes (usually 5 or 10) when you activate. Give them a false name and false address when you register the phone or buy one used. Give them a false area code if they ask. You will be assigned a number that has no connection to you.
8. Anonymous email service. One of the first things you'll need to register on Facebook is an email. You can do what most script kiddies do and make a fake Yahoo account, or do things the right way and sign up for Proton Mail. Proton Mail has servers outside of US jurisdiction (the Swiss, best of privacy for banking, now offers anonymous email and hosting). The mail service is free but I recommend donating to them because they offer a valuable service. They accept Bitcoin
Proton Mail also encrypts your data on the client side. So even if their servers get hacked or the US Government starts demanding data by threat of war or taking them off Special Drawing Rights, not even they can see what was sent or received.
If all this sounds a little over-kill, that's because it is. Your privacy on the internet is routinely disregarded and you have to take serious efforts to maintain your liberties now. Best of luck to you.
-Michael