Languages that satisfy the pumping lemma but aren't regular?
The language $\{ \$ a^nb^n \mid n \ge 1 \} \cup \{ \$^kw \mid k\neq 1, w\in \{a,b\}^* \}$ seems to be simple. The second part is regular (and can be pumped). The first part is nonregular, but can be pumped "into" the second part by choosing $\$$ to pump.

(added) Of course, this can be generalized to $\$L \cup \{ \$^k \mid k\neq 1 \} \cdot \{a,b\}^*$ for any $L\subseteq \{a,b\}^*$. Sometimes the formulation is in the "if ... then ..." style: if $w$ starts with a single $\$$ then it is of the form. That I personally find less intuitive.

As noted by @vonbrand the (possibly) non-regular part of the language is isolated by intersecting with $\$\{a,b\}^*$. This can be separately tested using the pumping lemma if needed.