Abarim Publications' online Biblical Greek Dictionary
The verb ασκεω (askeo) means to work or form artistically: to work raw materials into beautiful things, or even oneself, in which case it means to doll up, to adorn oneself. Our verb may describe the beautifying of existing buildings, or it may apply to deities, in which case it means to honor or do homage.
Specifically, our verb is used in the classics to describe athletic or gymnastic training: to train or practice. Later it came to be used to describe a broader practicing or training, but still with the idea of beautifying oneself in an artistic sense. It's used in Acts 24:16 only, and appears to relate to the following:
The noun ασκος (askos) describes a wineskin, or other such container made from leather. Despite the noted objections of modern experts, to the average Koine user, our noun looks identical to a noun derived from the previous verb — and would thus literally describe a beautifully trained thing, or a thing for beautifully preserving a liquid (and see our article on οινος, oinos, meaning wine).
In our article on the noun βυρσευς (burseus), a tanner, that is someone who makes leather from animal hides, we propose that, since texts were commonly kept in protective leather envelopes, our word ασκος (askos) also symbolizes whatever outer cover (the physical ink on paper) contains refreshing gulps of meaning within.
Our noun is used 12 times; see full concordance.