Gamble your way through!

Probability of getting at least one double six must be greater than $\frac{1}{2}$ . So probability of getting none double six must be less than $\frac{1}{2}$

$\dfrac{1}{2}>\Big (\dfrac{35}{36}\Big )^{r}$

Taking log both sides and transposing, $\frac{\log \frac{1}{2}}{\log \frac{35}{36}}<r$

The sign of inequality must be reversed as we are dividing by negative quantity $\log \frac{35}{36}$

By using calculator, $r>24.60$

So the minimum possible $r$ would be $\boxed{25}$