You have to escape $
by \
:
db.myCollection.find({ myKey : /.*\$aus.*/i });
// OR
db.myCollection.find({myKey: { $regex: '.*\\$aus.*', $options: 'i'}})
You can use this:
db.myCollection.find({myKey:{ $regex:new RegExp('^' + 'test$australia'.replace(/[-\/\\^$*+?.()|[\]{}]/g, '\\$&') + '$', 'i')}})