virtualbox.org

My host is Debian wheezy 64-bit running VirtualBox 4.3.2. I downloaded the Cinnamon .iso's for Linux Mint 15, both 32-bit and 64-bit. Neither will start in VirtualBox. With each of them, I can only get as far as the automatic boot message disappearing upon reaching 0 seconds. Once that happens, nothing. There is no longer any disk controller or CD activity after that point.

Post the guests log file (as an attachment)
See Online Users Manual for the location of the file.

log attached

I don't see anything that would cause the guest not to start. I do see some unusual locations for the guest, snapshots, Etc. but give you have selected to use them you must know about the associations and permissions. One thing you need to be aware of is if this is being stored on a USB device it will tak a while to do anything.

I have been using VirtualBox a long time, since (I think) 1.2. That pre-dates the $HOME/VirtualBox\ VMs directory structure. I use a separate hard drive for my guests. I have two Win32 guests running right now.

I copied the .iso from the USB 3.0 hard drive to the internal SATA drive and tried again. I even waited several minutes, but I still could not get any further than I did before. New log attached.

I see that you selected linux26 as the distro type. Maya (15) is based on Ubuntu raring (13.04).
Try selecting Ubuntu or ubuntu-64 depending on what you are installing.

Perryg wrote:Maya (15) is based on Ubuntu raring (13.04)

On a related side-note, where did you come up with this info? I've been trying to see which Mint-$(version) is based on which Ubuntu-$(version) but I couldn't... (btw, did you mean "Olivia"?)

Yes Olivia. Danged code names. anyway Mint 15 is Ubuntu 13.04. They have a Debian base LMDE as well.
I follow/test so many distros that I just know but distrowatch is a good place to find out about the distro.
They also follow the Ubuntu release cycle closely and usually come out with a new build about a month after Ubuntu releases, so Mint 16 will be Ubuntu 13.10, but the actual release number doesn't matter to VirtualBox, just that is is based off Ubuntu.

Setting Ubuntu64 fixed it and I got Mint installed
And I thought that setting was only for determining the icon to show in the VM list

The important bit is not 'Ubuntu' but '64'. To install a 64-bit guest you need to select a 64-bit guest OS type.