Regular Expression for the language that requires one symbol to occur at least once I am trying to figure out the simplest way to do this using a regular expression. 


Three symbols a, b, c.
The sequence length is unlimited, i.e. *.
The symbol a must be somewhere in the sequence at least once, but can appear more than once. 
The sequence may have only a.


More formally, $\{ w \in \{a,b,c\}^* ~|~ \#_a(w)\ge 1 \}$, where $\#_a(w)$ is the number
of $a$s in $w$. 

The best I get is


  $( ( b \mid c )^*\, a\, ( b \mid c )^* )^+$


Is that the simplest way?
The simplest regular expression I can think of for this is $\left(a \mid b \mid c\right)^* a \left(a \mid b \mid c\right)^*$. This is simpler than yours by the following measures of complexity:


It contains less nesting (and fewer parentheses in general)
It contains fewer quantifiers