UPDATE July 23, 2014: The impossible coin has been obtained!
The method used is similar to what is described in my "Attempt 2" below, with the crucial difference being the discovery that one could perform a jump immediately after exiting the water. The video was tool-assisted, but the method looks feasible to do in real time. Congratulations to pannenkoek2012 for doing the impossible after all these years; the images below date back to 2002.
Out of the 2092 coins in the 15 courses (excluding the new infinite coin glitches), there is only 1 coin that continues to defy the experts. This coin, dubbed "the impossible coin" is seemingly impossible to get because it is underneath the ground, just out of reach. It is located within the Tiny Island of Tiny-Huge Island, making 192 total coins in this course. The coin was most likely just overlooked by the game's programmers and wasn't supposed to be in the game. (It was probably intended to be placed above the ground instead of beneath it.) In fact, there is another (attainable) coin underneath the sloped ground beside the impossible coin. Although you cannot see this coin when you look at the slope, when you walk over the slope it is close enough to the ground that you can still pick it up. The impossible coin was discovered by Josiah on the GameFAQs Super Mario 64 board in January 2002 (along with a secret coin in Snowman's Land). Here are various pictures of the impossible coin...
The impossible coin is located just underneath Mario:
View 1
The impossible coin is the coin closest to the top of the image.
To see the coin from this view stand here.
View 2
The impossible coin is the coin farthest away.
To see the coin from this view stand here.
View 3
The impossible coin is the coin on the right.
You can see from this view it is just below the ground.
The other coin is close enough to the ground so that it can be obtained.
To see the coin from this view stand here.
View 4
The location of the impossible coin can easily be seen here:
View 5
The impossible coin is the coin on the right.
Once again, you can see how the coin is just underneath the ground.
This image was taken using a glitch to swim inside the island. (see below)
Close-Up View
Using some camera tricks, here is a close-up view of the coin:
Some Attempted Methods:
Attempt 1 - There is a trick that can be used to blast Mario right through the island, although not in exactly the right place. You can do the backwards long jump on the small beach, and by rapidly pressing A you can go right through the wall and through the island. Here's a quick screenshot I took of the trick (you can see the impossible coin in the top left corner). However, you can't get enough height or aim in the right direction, it seems.
Attempt 2 - My closest method so far is probably getting to the secret area beside the ! switch:
VIDEO OF THE FOLLOWING STEPS: Link
Step 1. Drain the water and complete the glitch that allows you to get inside the island. (Stand on the ball generator and double-jump towards the island peak and dive into the slope, sometimes you'll pass right through into the water inside). Then position yourself towards the secret area, like in this picture.
Step 2. Swim out of the water towards the secret area, and then dive once you start falling. (Unfortunately, you'll fall underneath the coin.) You should dive down a bit before leaving the water so that you don't go through the wall. Picture right before diving.
Step 3. You should land in the area underneath the piranha plant (another normally inaccessible area). Line yourself up with the secret area like in this shot.
Step 4. Jump out of the water (the previous shot shows where you should be before jumping) and try to land in the secret area. Usually you'll grab onto the edge, but once you pull yourself up (through the wall) into the area you'll usually fall back when you try to walk forwards. If you don't move at all you'll slide very slowly back. (Strange...) I find that with a quick jerk of the control stick you can get into the area (don't keep holding it down or you will run off the other side). Once you're in this area, you don't have much room to walk around in because you don't want to walk right through the wall to the ! switch. But you can get a great view of the impossible coin:
Because there isn't a lot of room, and there's only a thin space in the wall you can go through, I've never been able to make any kind of a good jump towards the coin. And even if you could, is the coin within reach? And could you land safely? (You would have to try and reach the water again). I've only posted this method here so that maybe someone can find a use for it...