I'm working with QGIS and have a big DEM raster layer with height information and a vector layer with many polygons, representing 1 x 1 degree tiles. Now I want to find for every polygon the highest value of the DEM pixels in it.
Try SAGA module Shapes - Grid -> Grid statistics for polygons - directly or via QGIS:
One of statistics is Maximum - so the result will meet your expectations.
Resulting table will be your input table with new columns named after statistics used in module (as far as I remeber SAGA doesn't obey the 8 character fieldnames limitation for shp - but you can edit it later).