I don't know that much about VPN, so it could be that this is utterly impossible. Here's hoping, though.
I would like to forward a port from my router to a machine connected to the network via VPN. That way, I could run a simple server on my laptop, and it would continue to work at the same address no matter where I am. The trick is that I don't want people connecting to the server to have to join the VPN. It should be completely transparent to the end-user.
If it's not possible with VPN, is there some other technology that would make this possible? Basically just forwarding a port to a remote machine, one without a static IP.
Yes, it's possible. You have to install the vpn on your router and make sure your vpn server always give the same ip to your laptop. (This setup is quite easy using Openvpn as server, but you need a router that has VPN capabilities)