My feeling is no - you are asking the same question this guy was asking but no-one could seem to grasp that he didn't mean 'can I run iTunes ON the PC but with the library on the external hard disk?' but instead was asking 'can I run iTunes installed on the external hard disk itself?'. He could not.
This implies you could do it if DH's work laptop was a Mac but I'm guess it isn't.
However, you can install just the iTunes application but have all the content (music, vids and stuff) on the external drive. IF DH is allowed to install software on his laptop.
Brief rundown: my macbook was stolen in a burglary last month and my iPod has died. I have a new iPod but the only computer we have is DH's work laptop and for obvious reasons I don't think it's a good idea to use it to run my iTunes account. I only used my macbook for iTunes as it was very elderly so I don't want to buy another laptop or a home PC.
So, my master plan is to buy an external hard drive to connect up to DH's laptop, install iTunes on it and store all my music on it so that the only thing I'm using DH's laptop for is to connect the external hard drive to the Internet and my iPod. Would that work or would I have to install iTunes on his laptop hard drive too?
If it would work, any recommendations for a cheap-ish external hard drive?