One possible approach is the following:
1. Set up your dashboard: drag Region and Country to Columns, and SUM(Count) to Text.
2. Create a calculated field, TotalByRegion, that is an LOD working on the Region level and ignoring the Country level:
Drag the TotalByRegion to Text to see that it works properly: all countries in Europe will have total of 6, and all countries in Asia - total of 3.
3. Create another calculated field, RankByRegion, as follows:
Using ATTR is safe, as the previous definition guarantees TotalByRegion is constant for every region.
4. Now, drag RankByRegion to filters, and choose 1 as the only allowed value. That guarantees we are choosing only the region with highest rank, i.e. highest total.
I'm attaching a workbook using the table you proposed above. Let me know if you have any questions.
MaxByRegion.twbx 6.0 KB
It is useful.