Question

I wonder if there exists one example of non-uniruled algebraic variety with level one Hodge structure.

Answer 1

I edited the answer to expand it and add more context:

The question was whether there exist nonuniruled smooth projective varieties with Hodge numbers $h^{pq}=0$ for all $|p-q|>1$. Of course, any curve of positive genus has this property. In dimension $2$, an Enriques surface, or any surface with $p_g=0$ and nonnegative Kodaira dimension, will work. Using Kunneth's formula, we can see also that by taking products of Enriques surfaces or products of such surfaces with a positive genus curve, we have an example in every dimension.