The Eggman appears directly north in Moongolow near one of the secret items if you have at least 5,000 golden eggs total and at least 1,000,000 gold. He acts similar to the Merchant, but only sells golden eggs.
When you touch the Eggman a shop will be opened with 2 options; buy 1 golden egg for 10,000 gold or buy "MAX EGGS". Each option can be purchased infinitely many times. Purchasing maximum eggs will always leave you with 0 gold and most likely a fraction of an egg (i.e. your egg count will have a decimal point.)
To remove your golden eggs or movement speed buffs, see Healer.