Learn Finnish: Kukaan ei etsi mitään paitsi tätä erää. - Nobody is looking for anything except this batch.

It is governed by the preposition paitsi ("except"), which requires the word that follows it to be in the partitive case. The main verb etsiä ("to look for") also happens to take a partitive object, but paitsi ensures the exception phrase stays in the partitive regardless of the main verb.

Why does Finnish use kukaan ei ... mitään (literally "nobody doesn't ... nothing")?

Finnish requires negative concord (double negatives). When you use negative pronouns like kukaan ("nobody") or mikään ("nothing/anything"), you must still include the negative verb ei. While English says "Nobody is looking for anything," Finnish pairs the negative subject (kukaan + ei) with the negative object (mitään).

Why is the subject kukaan in the nominative case, but the object mitään is in the partitive?

Kukaan is the subject of the sentence ("nobody"), so it takes the basic nominative case. Mitään is the object of the action. Because this is a negative sentence, the object must be in the partitive case. (Additionally, the verb etsiä inherently requires a partitive object even in positive sentences!)

What kind of "batch" does erä refer to in this context?

Erä is a highly versatile noun. It can mean a "batch" (like in manufacturing, brewing, or baking), a "lot" or "shipment" of goods, an "installment" of a payment, or even a "period" or "round" in sports. Here, it most likely refers to a specific physical batch of goods or a batch of data.