Obedient or attentive to an excessive or servile degree
Also, bootlicking: arselicking etc
When I hear the word obsequious it conjures up an image of a very large/portly in the extreme, greasy/oil covered man who is trying to get something from someone by fawning all over them. This man (to me) looks remarkably like the Baron from Dune!
late 15c., "prompt to serve," from the Lain - obsequiosus "compliant, obedient," from obsequium "compliance, dutiful service," from obsequi "to accommodate oneself to the will of another," from ob "after"+ sequi "follow" (see sequel). Pejorative sense of "fawning, sycophantic" had emerged by 1590s.