Polyptoton

(Definitions)

"...had, having, and in quest to have, extreme..." -- Shakespeare, Sonnet 129

What is the term for a repeated word or phrase in the same sentence?

Polyptoton, from the Greek, "multiple cases", "many inflections".

Would the auctioneer's chant "Going,going,gone!" be another
example?

Fit casibus modo hoc schema, quod polyptoton vocant. / This figure may also be effected solely by change of cases, a proceeding which the Greeks call polyptoton. [Quintillianus IX.iii.37] From ptosis 'falling, (grammatical) case' from pipto 'to fall'.