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Khoth comments on Open Thread, April 15-30, 2013 - Less Wrong
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0 doesn't have an inverse in the hyperreal numbers either (To see why this it true, consider the first-order statement "∀x, x*0 != 1" which is true in the real numbers and therefore also true in the hyperreals by the transfer principle). From this it obviously follows that you can't multiply a number by the inverse of 0.
Further, if you did decide to adjoin an inverse of zero to the hyperreals, the result would be the zero ring.
Going to have to investigate more, but that looks solid.