Learn Turkish: Aktardan demet demet taze nane alayım. - Let me buy bunches of fresh mint from the herbalist.

This is an ikileme (reduplication). Repeating a measure word or noun like demet (bunch) emphasizes abundance. Instead of just saying "a bunch" (bir demet) or using the standard plural (demetler), demet demet vividly expresses "bunches and bunches of" or "in large quantities of bunches".

Why use alayım instead of a regular future or present tense verb?

alayım uses the optative mood (-a/e plus the personal ending), meaning "let me buy" or "I should buy." It's used for suggestions, wishes, or immediate intentions. If you used the future tense (alacağım), it would sound more like a stated fact ("I will buy") rather than a spontaneous thought.

Does nane need the accusative case here since it's the object being bought?

No, because it is an indefinite object. The speaker is buying "fresh mint" in general, not a specific batch of mint that was already being talked about. If you wanted to say "Let me buy the fresh mint," you would add the accusative ending: taze naneyi alayım.

What exactly is an aktar? Is it just another word for a pharmacy?

No, an aktar is a traditional Turkish spice seller or herbalist shop. They sell dried herbs (like the mint in this sentence), spices, loose teas, and natural remedies. A modern pharmacy that sells prescription medicine is called an eczane.