Is there a shape with infinite area but finite perimeter? Is this really possible? Is there any other example of this other than the Koch Snowflake? If so can you prove that example to be true?

There isn't—on a plane. However we may take a 3D cylindrical coordinate system and define a surface $z=\sqrt r \sin\tfrac 1r$. The $\tfrac 1r$ term makes infinite sequence of waves towards the center, $\sqrt r$ term reduces their height (amplitude) so that the surface has no singularity there.
Then we cut a part of the surface, containing the center point – it will possibly have an infinite area.
The function above is just a sketch, I didn't check if the area is actually infinite. In case I'm wrong you might either increase the waves' frequency growth, say by replacing $\frac 1r$ with $\frac 1{r^3}$ or with $\exp(\tfrac 1r)$, or take the steeper amplitude term, say $\sqrt[4] r$ instead of $\sqrt r$.
