ABC Conjecture. For every $\varepsilon>0$, there exist only finitely many triples (a, b, c) of positive coprime integers, with a + b = c, such that:
$$c\gt\operatorname{rad}(abc)^{1+\varepsilon}$$
Obviously we are trying to get $\varepsilon$ as small as possible, but what if we let $\varepsilon=1$.
As (conjecturally) there are only a finite number of solutions, what is the maximum value of $c$ for $\varepsilon=1$?
Or, if this is still too hard, find $C_{max}$ for integer $\varepsilon$'s.
The article you link to goes on to define the quality of an $abc$-triple, to be the number $q$ such that $$ c=\operatorname{rad}(abc)^q $$ So looking for triples that appear when $\epsilon=1$ is tantamount to looking for triples whose quality is greater than $2$.
The triple with the largest-known quality is said to be: $$ a=2\\ b=3^{10} \times 109\\ c=23^5 $$ which has a quality of about $1.6299$. That is, there are no known triples which satisfy the statement with $\epsilon=1$, or indeed for any $\epsilon\geq0.63$.