I don't think there is a good solution in the general case. The closest I can think of is Charts Menu -> Statistics -> Family tab, which includes "top 10" lists of couples with the largest number of children, and largest number of grandchildren. Interesting, but no help for an arbitrary individual, and it also doesn't report on great-grandchildren.
Beyond that, everything I can think of involves manual intervention. Personally, I would try either Charts Menu -> Interactive Tree (where you must count the number of boxes in a column), or Reports Menu -> Descendants (where you must count the number of children among people of the previous generation)
1) The number of people in the first generation is 1: just the proband
2) menu Charts - Descendants: select 2 generations with layout Individuals. The total number of people is shown at the top (showing x of y), so the number of children of the proband is this number minus 1.
3) On the same chart select 3 generations. Take the total and subtract the total you had at 2). This is the number of grandchildren.
4) Select 4 generations. Take the total and subtract the total it showed at 3). This is the number of great-grandchildren.
If you use v1.7 then don't use this method since also partners are included in the list and thus get counted.