* ὀρέγομαι (oregomai): This verb means "to reach out for," "to strive for," or "to long for." It carries a sense of reaching towards something desired, and is probably the closest single word.
* ἐπιθυμέω (epithyméo): This means "to desire," "to yearn for," or "to long for." It emphasizes the desire itself more than the active striving.
* Words incorporating the concept of ambition (φιλοδοξία - *philodoxia*) or ambition's aim (στόχος - *stochos*) could also be used in a phrase to convey the meaning of "aspire."
The best choice would depend on the specific context of how "aspire" is used in the sentence.