To my knowledge, without swap you can't use Hibernation. So that's one type of applications that may crash if you don't have a swap file or partition. I'm also confidently sure that activating certain options on VirtualBox without virtual memory on the Host machine will eventually crash the guest machine.
However, to be honest I have an hard time understanding why, with the exception of physical space limitations, would anyone wish to run their system without a swap partition or a swap file. Many programs can't deal well with Out Of Memory exceptions. How many of us here coded our programs to end gracefully if a OOM exception shows up? I'd guess few or none of us.
What's worse, contrary to many systems, Linux overcommits memory. The way the kernel deals with this is to launch OOM Killer that will simply kill existing processes, which may invariably destabilize the system if the wrong processes are chosen to be terminated. Which inevitably will happen if the situation persists and OOM Killer keeps getting triggered. And it will keep getting triggered, if you don't have the concept of virtual memory in your system that will facilitate disposing of memory pages that are no longer needed from running processes.
I run on 4GB and only very rarely I see my swap file being used. (I always preferred swap files to partitions). So I can imagine how annoying it can be for someone running a non-server computer on 12GB and wasting disk space on a swap file or partition. But you do it for stability. Besides, if you don't plan to ever hibernate, all you need is maybe 2 or 4 GB of your disk capacity on a 12GB RAM system.