Hosting Laravel app

Laravel Forge does not host a web application itself; it provisions a server on a hosting service (or BYO server).

You can get going on DO/Linode relatively cheaply (depending on your location); Forge is additional.

Alternatively, AWS and Heroku have free and hobby tiers to get you started.

Also @demonz prior to choosing, understand how the free tier works:

https://docs.aws.amazon.com/awsaccountbilling/latest/aboutv2/billing-free-tier.html

Lookup Heroku as well. Also if you choose a shared host instead, go for one of the paid plans, because the free plans are very limited and in many cases you can't even setup laravel correctly.