Learn Finnish: Me teimme tuon hauskan kepposen hänen huomaamattaan. - We did that funny prank without him noticing.

It uses the 3rd infinitive abessive form of the verb, which expresses doing something 'without' an action. The root verb is huomata (to notice). We take the stem huomaa-, add the abessive ending -matta, and then attach a possessive suffix -an to match the genitive subject hänen.

Why does the word end in -taan instead of using the standard third-person possessive suffix -nsa?

In Finnish, the third-person possessive suffix can be either -nsa/-nsä or a lengthening of the preceding vowel plus -n (Vn). Both huomaamattansa and huomaamattaan are grammatically correct, but the vowel-lengthening version (-taan) is much more common and natural in modern Finnish.

Could we drop the word hänen since the possessive suffix already tells us it is a third person?

No, you need hänen here. If you omit the genitive pronoun, a possessive suffix automatically refers back to the subject of the main clause. Since the main subject is me (we), dropping hänen would imply 'we did the prank without us noticing'. You must include hänen to show that someone else failed to notice.

Why is the phrase tuon hauskan kepposen in the genitive-looking accusative case?