Essential Equipment
(in order of importance)

• Passport and race ticket.
• Cash (Euros - bah, I remember when it was Francs) - for the fairground, sandwiches, t-shirts, etc. (£70 should be enough to keep you in junk food, multiple goes on the dodgems and get you a T-shirt).
• If you're driving, take a colour copy of your driving licence, you don't want the Gendarmes taking your original (as they can).  You don't want to spend 6 months chasing around to get it back.  Put a copy of your passport in for good measure too.
• Sunblock (though a brolly could be useful just incase!) and sunglasses.
• Sleeping bag - unless you're Dave Tyrer, then a pair of woolly gloves will suffice (WTF?!).
• Cheap lager from the ferry/Calais.
• A bag for lugging afore-mentioned lager around the circuit
• A radio to find out what's actually happening on the track! Radio Le Mans transmits on FM and provides invaluable coverage of the race.
• Bog roll, should your bowels not be as disciplined as they ought to be, or your body can no longer take the toxicity of what you need to coil out.
• Change of clothes, more optional than essential if it's dry.
• A tent, if you're a softy - also very useful if it pisses down in the night as it did severely in '98. Beware though, camping space can be like gold dust.  If you're camping one of those blow up mattresses can be a Godsend.
• A hat, but they're always giving away peaked caps should you forget one.
• A camera (spare films & battery if required), should you wish to pictorially record any of it!
• A flag - with as tall a pole as possible.
• The Autosport magazine Le Mans supplement, usually available from the newsagent during the week of the race.
• Earplugs, this depends a lot on where you sleep, but LM during race weekend is LOUD.  If you're camping don't underestimate the non-stop euro discos that go on all night and the fireworks that literally make the ground beneath you shake.
   

Notable Exceptions to the above

The Passport rule doesn't strictly apply if you're Sniffer, it's more a desirable optional extra in his case. For everyone else I would heartily recommend you do bring along your passport.  Whilst in France do attempt to keep your passport, though again if you're Sniffer this is more desirable than essential.

Dave Tyrer's policy of not taking a sleeping bag, but ensuring you do pack some woolly gloves is not recommended.  Even though the race is in June the nights do get chilly, while a pair of woolly gloves will provide ample protection for your hands, the rest of you will freeze.