Learn Polish: Nigdy nie widziałem takich tajemniczych sztuczek! - I have never seen such mysterious tricks!

This is a great example of Polish end-focus. Because Polish word order is flexible, the very end of a sentence is the prime real estate for the most important, new, or emphasized information. Placing the noun phrase at the absolute end highlights the speaker's sheer amazement at the tricks themselves. If you rearranged it to Takich tajemniczych sztuczek nigdy nie widziałem, the emphasis would shift away from the tricks and onto the fact that you have never seen them.

Why is "takich tajemniczych sztuczek" in the genitive case here?

It is in the genitive case because of the negated verb nie widziałem. The verb widzieć (to see) normally takes a direct object in the accusative case. However, Polish has a strict rule called the genitive of negation: whenever a transitive verb is negated with nie, its direct object must switch from the accusative to the genitive case.

What is the base form of "sztuczek", and why does it have an "e" in this form?

The nominative singular form is sztuczka (a trick or stunt). To form the genitive plural of feminine nouns ending in -a, you normally just drop the final -a. But doing that here would leave an unpronounceable consonant cluster (sztuczk). To fix this, Polish inserts a "mobile e" between the final consonants, giving us sztuczek.

The sentence uses both "nigdy" and "nie". Is a double negative always required?