request to officially ask for something
consult to ask someone for advice
demand to ask for something very forcefully
nag/pester to keep asking for something many times, in an annoying way
beg/plead to ask for something in an anxious or urgent way, because you want it very much
question/interrogate to ask someone questions - used especially about the police or the army
grill informal to ask someone a lot of difficult questions
cross-examine to ask someone questions in court - used about lawyers
poll to ask a lot of people for their opinion about something