Regex for the language over $\{a,b,c\}$ that contain at least two $a$'s but no two consecutive $b$'s
You'll need to guarantee the following:


at some place, you have $aa$
whenever there is a $b$, then there is a different letter between it and any further $b$


So one part of your regular expression will be $aa$, and that'll be need to be "in the middle" since we don't want to exclude the word $bcaab$ for example. The other part needs to take care of the restriction for $b$s.

One way to make ensure the restriction with regard to $b$ is to put at least one letter before every $b$ except for the first $b$ before and the first $b$ after $aa$. So you could do:

$(b + \epsilon)((a+c)^+b)^*(a+c)^*aa(b + \epsilon)((a+c)^+b)^*(a+c)^*$

where $\epsilon$ is the empty word. If you don't want to use $\epsilon$, you can split that expression up into 4 expressions without epsilon and sum them up.