The remainder when 6^666..∞ is divided by 10 is :

$$\eqalign{ & {\text{Since}}, \cr & \frac{6}{{10}} \to \,{\text{remainder}}\,{\text{is}}\,{\text{6}}{\text{}} \cr & \frac{{{6^6}}}{{10}} \to \,{\text{remainder}}\,{\text{is}}\,{\text{6}}{\text{}} \cr & \frac{{{6^{{6^6}}}}}{{10}} \to \,{\text{remainder}}\,{\text{is}}\,{\text{6}}{\text{}} \cr & \frac{{{6^{{6^{{6^6}}}}}}}{{10}} \to \,{\text{remainder}}\,{\text{is}}\,{\text{6}}{\text{}} \cr & \frac{{{6^{{6^{{6^{{6^6}}}}}}}}}{{10}} \to \,{\text{remainder}}\,{\text{is}}\,{\text{6}}\,{\text{and}}\,{\text{so}}\,{\text{on}}. \cr & \cr & {\text{Thus,}}\,{\text{in}}\,{\text{all}}\,{\text{the}}\,{\text{such}}\,{\text{cases}}\, \cr & {\text{The}}\,{\text{remainder}}\,{\text{will}}\,{\text{always}}\,{\text{be}}\,6 \cr} $$