WARNING: This page contains SPOILERS for Final Fantasy VII.
After taking the Ropeway up to the Gold Saucer, you will be in an area with the ticket lady and a save point. Above that save point (and to the right) is a house. On the left side of that house is a small area that appears to be the door. Occasionally, a man will appear there and offer to sell you GP. In this section, I will list everything I know about this GP Dude.
When will he appear?
So far, I've done thousands and thousands of tests on this issue. In more than a thousand trips into the area, the GP Dude didn't show up once pre-Bronco. In more than a thousand trips post-Bronco, he still did not show up once. In the two thousandish trips I've made post-Highwind, he appears about once in every seven trips into the area. So, I theorize that he will not show up at all until you get the Highwind. Although I admit that I have not yet tested it on disc two pre-Highwind. I'll get to it one day, and I'll make sure to update the results.
Where is he?
How do I make him appear?
The easiest way is to have a ticket to the Saucer. You don't have to go up and down the Ropeway. Simply enter and exit the Terminal Floor (the gold room with all of the pipes). Keep your eyes on the spot where the GP Dude appears, and he will eventually show up.
How often will he appear?
An average of one time for every seven times that you enter the area. Yeah, that's right, once in every seven. A lot of people have no patience, and they severely overestimate his rarity. So far, my records are twenty-two consecutive non-appearances and eight consecutive appearances.
How much GP can I get from him?
He will sell you One GP for 100 gil, with a limit of 100 GP at each appearance. You can keep returning to him to buy more and more chunks of 100 GP though. I say this because I have seen some FAQs and board users claim that you can only buy GP from him once. That's outrageously false.
Is this the fastest way to get GP?
Yes. Assuming you've got the Highwind, anyway. Chocobo racing is much more random as far as GP returns from each race, and the races take longer than it takes to make seven trips into the GP Dude's area. Not to mention the fact that you have to have a good chocobo in order to get decent amounts of GP from each race. But there are some good items to win in the races, so I highly suggest going there first. In the process of racing your chocobos while breeding a Gold one, you're likely to make all of the GP you'll need for the rest of the game.
Props to shadow for providing yet another picture. You should see what I was using before I got this... >_>
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