Is there any particular reason the A380 doesn't have a centre wing tank? Today I learnt something new, but apparently the A380 has no centre wing tank, which leads me to ask, is there a specific, design reason for this? Or did Airbus just think those huge wings were big enough to hold all the fuel they'd need?

Although, I've heard rumours that the A380F if built would have had a centre tank (no idea how true this is), but this shows the engineering is possible, Airbus just chose not to go with it for the passenger version. Any reason why?
It simply doesn't need one. Depending on the source, the A380 has a range from 8200 NM1 to 8477 NM2. The current longest scheduled flight is AKL-DOH, which is 7848 NM3. So the current configuration is capable of flying any of the current routes. 

As far as the A380F, it really wouldn't make sense to add a center tank there either. The extra fuel weight would mean less cargo capacity available.

 1 Wikipedia

 2 Aerospace Technology, 11/19/2013

 3 Wikipedia