Why does it seem as if I can apply the Pumping lemma to a language that is regular?
The pumping lemma says that, for every regular language there exists some pumping length&nbsp;$p$* such that blah blah blah.

You made two mistakes. First, you tried to choose&nbsp;$p$ based on a specific word, rather than based on the language as a whole. Second, you chose the wrong value of&nbsp;$p$. In particular, for the language you've given, $p=1$ has the property required of the pumping lemma: for every word&nbsp;$w$ in that language, you can write $w=\epsilon y z$ with $|y|=1$, and you'll find that $\epsilon y^n z$ is in the language for all&nbsp;$n\geq 0$.