Do not forget your photo identification, this cannot be stressed enough, you won't be able to pick up your event number if you don't have this, I'm not sure if they accept anything other than passport or driving license, you might want to investigate that, if you don't have either of those.
Clothing (during the event)
A pair of running shoes that you don't mind getting soaked, covered in mud and full of tiny little bits of grit, I wear a pair of trail shoes that have a wear just for events addidas kanadia trail shoes, which have a tough but flexible bit of plastic to protect toes from knocks. They are nice and light to run in until the grip gets clogged with long grass and mud but they are easy to clear the soles if needs be.
This is a relatively tight fitting lycra top that acts like a wetsuit heating up a layer of water around you and will help you to stay warm after you've been dunked in freezing water. You will be cold in the water but you won't stay cold for too long after you've left the water so this is a real help on the days when its raining or hail is coming (yes it did hail at London West in 2014) you should be able to get one for around £20 and is worth investing in.
Wicking running top,
Being as dry as possible is a must when you aren't in the water obstacles, this bit of kit will likely be made of polyester which won't hold the water and will dry out which will help keep your body temperature up during the times when the air temperature drops.
These should be something which won't hold the water and allow freedom of movement as running in shorts weighed down with water is tiring, which combined with the course will wear you out in no time.
My dad wore a pair of these as he didn't want his legs scratched to hell, you'll need some that are reasonably thick given that the course will have sections that will wear away and thin material and leave you a little exposed.
I wear a pair of fingerless weight lifting gloves which are perfect, they protect my palms when helping people over obstacles, they can be an issue when it comes to holding onto bars or rings but try and find some that will dry as you're running.
Whatever feels most comfortable, I have some short running socks that have padding in the heels and arches.
There are a number of water stations dotted around the course, to provide entrants with a much needed drink but I like to take water with me, although it is a bit heavier when you start having water with you is helpful, especially on the hotter days. These are about £35 for the 2 litre classic that I have, which has a zip pocket for storing things in. Camelbak
Clothing (post event)
This is a must unless its a really warm day when you finish, since you'll need to get dried after the lovely cold shower they provide to entrants.
Unless you like wearing wet muddy clothes, you'll need something to change into, these will probably be the clothes you wore to get to the event, so make sure your bag is big enough to carry them.
Energy gels (optional)
Like the camelbak these are totally optional but I like to have them to hand to give me a little energy boost, they do offer you protein bars/bananas at some of the water stations but if slightly dry protein bars aren't your thing then pick up a couple of these might be useful, there are plenty of choices available and they normally cost between £1-1.50 so find a brand that you like the taste of as some of them aren't my choice. I did find the vooma energy gel from USN were quite nice but they seem to be unavilable individually now.
Energy bars (optional)
Similar to the gels with a range of flavours available, again find one you like the taste of and use that.
There are a number of vans selling all manner of food at the Mudder village but as usual with events like this I find they are a bit on the pricey side, so bring yourself something to eat on the journey home to replenish yourself, something with protein in it which should keep you from getting the energy spike from processeed sugars and give a longer energy release.
So that's my advice for the day of the event in a bit more detail, this is what I do whenever I take part, the food is quite important as you may be totally drained of energy when you finish.