At the same time?
Preferably hosting multiple devices, either via multiple ports or a USB Hub.
I'd like to program it to host both a mouse and keyboard, while being a slave to a PS3 pretending to be an HID compliant game controller. (example use, there are others)
Have a look at the Vinculum chip, made by FTDI.
This is an Arduino stuck on the end of an HID USB interface:-
http://www.create.ucsb.edu/~dano/CUI/
There are now cheap Microchip PIC micros with USB host controllers, and there are a variety of cheap micros with USB device controllers, so it may be the easiest solution to use one of each as peripherals to the arduino. I think most of the chips that can run both simultaneously are large and relatively expensive (The Viniculum someone mentioned is a 256k/16k microcontoller!) (OTOH, that "relatively expensive" keeps getting cheaper...)
Note that you have to be careful about "USB HOST" controllers, since for many of the USB peripherals that you might want to use, the host-side processing that needs to occur can be well beyond the capabilities of small micros. I talked to microchip at ESC a couple weeks ago about the issue, and they admitted that a lot of host-side drivers (ie for windows) were so large and complex that it was pretty impossible to talk to the devices (I was asking about BT and 802.11 dongles; they were complaining about even printers...) But your application (converting keyboard/mouse to game controller) seems like it ought to be possible.