Enterprising software engineers have recently found a way into the Wii U's file system through an exploit with the consoles browser. However, the particular entry point has reportedly already been closed inadvertently by to Nintendo's recent version 5.0 firmware update (via GI.biz):

"The bug is still in the browser, but when they added the quick boot menu in 5.0.0, it moved a bunch of code around and broke our ROP (Return Oriented Programming) chain that allows us to gain code execution," a post from the hacking group on the GBAtemp forum reads. "So it IS possible to get it working, we just need to find where the addresses are, which is difficult if you don't have any way to see (with the earlier versions we had binaries and so we could see where the code was), so it may take some time.

"This doesn't work on 5.0.0 since they changed stuff up on us, so first we'd need to port it to 5.0.0, and then our main priority is to start disassembling and looking for an exploit in the kernel which will allow us to do a lot more and allow for homebrew.

Although accessing the Wii U filesystem does not allow running unsigned code on the device, it can help pave the way for the kind of kernal exploits that do.

That said, that limitation hasn't stopped tinkering modders from playing around with game assets to alter the experience of Wii U games, as the below video of Mario Kart 8 demonstrates a couple of crude hacks including altering race music and changing course names.