Dimension of fibre products of $k$-schemes for an arbitrary field $k$
When you leave the realm of finiteness hypotheses, anything can happen.
Theorem (Grothendieck-Sharp): Let $L/k$ and $K/k$ be two field extensions. Then $$\dim_{Krull} L\otimes_k K = \min(\operatorname{trdeg} L/k,\operatorname{trdeg} K/k).$$
This gives you all the examples you want: just let $L=K=k(\{t_i\}_{i\in I})$, so that $L\otimes_k K$ has Krull dimension $|I|$ despite $L$ and $K$ having Krull dimension zero.