In a way, it is Exeggcute and not Wailord which are the largest of pokemon, for Exeggcute possess a unique ability to separate into their component eggs and spread out over vast distances while still functioning (through telepathy) as a single pokemon.
It was once believed that the reason for these searches was to find food, but Exeggcute possess more than enough food in their shells to last until even the last of them evolve. When trained, they are occasionally used in this manner for eavesdropping and espionage, albeit less for espionage than in the old days; a lone Exeggcute is typically mistaken for an egg, but those with sensitive information have learned to be suspicious of eggs.
Most say what they are seeking are leaf stones. Only a few realize that what they are truly searching for is a way to hatch like real eggs and become something other than Exeggutor. For although Exeggutor is a tolerable or enjoyable form for the heads, all fear that they will instead die to form the body, and none can know until evolution which will be their fate.