Drug having endless high hopes (7)

I believe the answer is:

aspirin

I'm a little stuck... Click here to teach me more about this clue!

'drug' is the definition.
(aspirin is a kind of drug)

I don't understand how the remainder of the clue works.

(Other definitions for aspirin that I've seen before include "Anti-inflammatory analgesic" , "Common painkiller" , "Headache killer?" , "Analgesic over in Paris" , "Anodyne" .)

I've seen this clue in the Sydney Morning Herald.

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