From what I have read, this all known examples of exchange rings that are not clean are based on this one. The description I am giving here is adapted slightly from Corner rings of clean rings need not be clean. by Janez S̆ter, Communications in Algebra, 40:5, (2012) 1595-1604, DOI: 10.1080/00927872.2011.551901
S̆ter comments that the construction below is actually isomorphic to the opposite ring of the ring originally described by Bergman.
Let $F$ be a field of characteristic $\neq 2$, and let $S=\mathbb{CF}_\mathbb N(F)$ be the ring of countably infinite column-finite matrices over $F$. (This is a full linear ring of transformations on a countably infinite dimensional vector space.) Let $R$ be the following subring of $S$: $$ \left\{[a_{i,j}]\in S \middle|\, \exists n_A\in\mathbb N \text{ such that } a_{i,j}=a_{i+1, j+1}\,\forall i\geq n_A, j\geq 1\right\} $$
In other words, starting from the top, there can be finitely many arbitrary rows, then a row $[a_1, a_2,\ldots]$ followed by $[0, a_1, a_2,\ldots]$ and then $[0,0,a_1, a_2,\ldots]$ and so on for all the remaining rows.