How can I quickly gain XP in Fornite?

There are many ways to earn experience in Fortnite. However, the general consensus for earning it quickly is to complete as many of the challenges as you possibly can. This is by far the fastest way to earn experience. The experience you earn can and will vary from challenge to challenge, with the low end around 1.5k and I've seen it go as high as 50k.

You'll also gain experience for generally, everything you do in Fortnite, by earning medals for basic actions such as the following, to name a few:

• Crafting Weapons
• Opening Chests
• Harvesting Building Supplies
• Building
• Hunting Wildlife

With this in mind, there are some other things you can do to speed the process up a little:

• Get Battle Pass: Once you have the battle pass, you have access to all challenges, exclusive events with new challenges, and themed rewards.
• Play with Friends: If you have the battle pass, all the friends you play with will get experience bonuses, and the bonus increases as you go up in level. You reap this benefit too when playing with friends that also have battle pass.
• Double XP Weekends: There are occasionally weekends with double experience events, so keep an eye out for when these are occurring and be sure to get some time in during those.

_{This answer was written using personal experience and various sources around the web (here, here, here, and here, are good examples). While improbable, this information could become dated with time. Feel free to edit my answer to ensure it reflects the latest information for future readers.}

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Hope this helps out a bit.