It is impolite to ask, "Can I bring [name of additional guest here]?"
If you are indeed wondering if you can bring your charming friend, mother, dog, etc., word it something like this:
"We'd love to come, but my mother will be visiting that Saturday. "
"Your party sounds great, but we've got Jack and Jill visiting from Europe."
Then be quiet and see what the host says.
This guideline applies across the board for all types of parties -- including weddings. If your children were not invited, do not bring them. If your significant other was not on the invite, it means she/he was not invited. Note that these exclusions, 90% of the time, are not personal.
You can not and should not know a host's reason for limiting invitees. It may be monetary, there may be a certain group dynamic going on, she may need exactly 10 for her Human Rope activity, or perhaps a surprise intervention is planned.
Asking to bring extra guests puts the host in an awkward position: she has to find a way to tell you "no" without pissing you off and creating extra tension when you (hopefully) do show up at her shindig. She's afraid of alienating you, but also wants to keep her party dynamics the way she planned.
And never, ever, ever ask if you can bring your pets. This creates the same awkward situation, and adds one more responsibility to the host's plate. Yes, I know your pooch is a little darling who won't disturb a soul. That's not the reality, and it's not your house.
It is okay to bring along others is if the party notice arrived via eVite, and the "invite other people" feature is turned on, or if the host (or invitation) specifically said, "guests/dates/children, etc. welcome."
Host's house=host's rules. Respect them. If it's too much for you attend without your mom/out-of-town friend/significant other/cat/landlord, etc., then don't attend.