If I clip PROBAV (https://zenodo.org/records/3939050/files/PROBAV_LC100_global_v3.0.1_2019-nrt_Discrete-Classification-map_EPSG-4326.tif?download=1) worldwide raster with Deflate compression setting and Save resolution setting the result is bigger than 6 gb. Although the source is 1.8 gb.
What other parameter(s) would make it smaller?
UPD: I used parameters mentioned in the answer and I got 756 Mb raster for Eurasia. Solution found.
gdalinfo clipped_raster
and also specify exactly how you clipped it.