Because rasterization works by projection to a plane, 180 degrees is the (already breaking) maximum.
Also, for a non planar projection like sphere maps you would need ability to draw curved triangle edges. To do this on GPU it would work to tessellate the edges so they can form curves. I remember to have seen this in some GPU gems book, and they did this for GI using sphere maps. For some reason i can't find the article anymore, but i would not consider this to be very practical. (NV hosts the books online for free)
A fast way to generate many small sphere maps was shown in the Many LODs paper, using point hierarchies and splatting. (https://perso.telecom-paristech.fr/boubek/papers/ManyLoDs/ManyLoDs.pdf)
The easiest way is probably rasterizing 5 faces of a cube map. Only front face needs full resolution line 512x512, the other faces (left, right, top, bottom) would require only 128x512, because sphere map only covers a half space.
There is no distorition when projecting resulting cube map faces to sphere map, but varying sampling rate between the two needs some attention for best results.
However, using just the cube map as is should work as well.