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Search for tiny planet photoshop - you'll find tons of tutorials. You want to do the opposite of the planet to get this effect.
Filter > Distort > Polar Coordinates, using Rectangular to Polar. The original should be seamless left and right.
The Polar to Rectangular option is the inverse. Using this, you can see the original of your example would look something like this:
The ball and sky would be on their own layers.
The following is pretty crude but it might give you some ideas.
Apply the Polar Coordinates filter from the Distort section of the Filter menu. Use the Rectangular to Polar option.
Resize the layer to crop the crappy corners, fix the seam at the top using something like the Spot Healing Brush and then darken the corners using something like a Curves adjustment layer.