There are a few examples that are finitely generated. 

(1) Let $L$ be Lyndon's groupoid given by the following multiplication table:
\begin{array} [c]{c|ccccccc} 
L & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 
2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 
3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 
4 & 0 & 4 & 5 & 6 & 0 & 0 & 0 \\ 
5 & 0 & 5 & 5 & 5 & 0 & 0 & 0 \\ 
6 & 0 & 6 & 6 & 6 & 0 & 0 & 0 \\ 
\end{array}
Then the variety $\mathrm{var}\, L$ is non-finitely based and an explicit basis is:
\begin{align}
(xx)y = x(yz) = zz, \quad
(\cdots((xy_1) y_2) \cdots) y_k = ((\cdots((xy_1) y_2) \cdots) y_k) y_1, \\ 
((\cdots(x_1 x_2) \cdots )x_k )x_1 = zz, \quad k=1,2,\ldots 
\end{align}

(2) Let $A_2$ be the 0-simple semigroup
$$
\langle a,b \mid a^2=aba=a,\ bab=b,\ b^2=0\rangle
$$
of order five and let $\mathbb{Z}_n$ be the cyclic group of order $n$. Then for each $n \geq 2$, the variety $\mathrm{var} \{A_2,\mathbb{Z}_n\}$ is non-finitely based and an explicit basis is:
\begin{align}
(xy)z=x(yz), \quad x^2 = x^{n + 2}, \quad xyx = x (yx)^{n + 1}, \quad
xyxzx = xzxyx, \\ (x_1^n x_2^n \cdots
x_k^n)^{3} = (x_1^n x_2^n \cdots
x_k^n)^{2}, \quad k=2,3,\ldots 
\end{align}