At one point in the late 1980's Microsoft had a GREATER than 100% market share of the Macintosh spreadsheet market.
How is this possible?
Market share (for a given period) is the participant's sales in the market divided by total sales. It just so happened that Lotus had more returns than sales of their failed spreadsheet, Lotus Jazz. So Lotus, had a negative market share and Microsoft had more sales of Excel than total sales in the market, resulting in a greater than 100% market share.
I don't remember the exact numbers and I believe there was at least one other competitor in the study. But let's just say the numbers were:
Microsoft: 102% Lotus: -2%
In that case the Herfindahl–Hirschman Index would be 102^2 + (-2)^2 = 10404 + 4 = 10408.
So, in this pathological case it is possible for the HHI to exceed 10,000.
Edited: Added (for a given period) above, for clarity.