Can I install a distro booted from a drive to the same drive?

Question

I know how stupid this sounds BUT is it possible to install a linux distro that has been loaded to the drive it was booted from?

I was thinking of installing Arch or Puppy to a flashdrive that Would be hosting itself.

OR

Is there a precompiled drag and drop style distro that you can drop the files to and run with constant persistence from drive to drive?

Other limiting factor is that the overall size of distro must be <4GB.

EDIT: To clarify, I want to install a system to a flashdrive that it is livecd'd from. Scenario for reference:

1. I use LiLi to put ubuntu-minimal.iso on a flashdrive
2. Boot ubuntu-minimal.iso from flashdrive.
3. Install the iso to the same drive it booted from.

Answer 1

I apologize for bringing up this question again, but I figured I should assign a proper answer with what I have learned.

You can install from disk, but the installation can't repartition the disk or overwrite the partition containing the image from which you install. – vonbrand Jan 24 at 16:24

To elaborate to anyone that this doesn't answer the question for,

You cannot install to the medium from which the installer is located as it would require the need to repartition the drive into a proper format. Eventually the system would destroy the files required to continue the installation.