I chose a certain sequencing rule, and I told my
friend no matter what number he named, it would
eventually end up at repeating 4s.

At first he chose 5, which ended in 4, 4, 4... Then he
tried 18, which met the same fate. Even numbers like
-42 and Pi ended in 4, 4, 4...

Still trying to outdo me, he started to get really creative. He tried 63424563562324 ... same. He even tried a googolplex and Avogadro's number. But alas all the numbers he tried ended up in 4,4,4. Finally he conceded.

What's the sequencing rule I use? Should he have conceded? Can you find any starting terms which don't end in 4, 4, 4...?