I know that it's in the company's best interest to lowball you: Is there a way to get around this without burning bridges? I'm wary of doing something like getting a counteroffer, since that feels very aggressive.
Adding on to this: Even though we're a startup, we do have a fairly defined career matrix. We have levels (junior, intermediate, senior, etc) and a performance review every 6 months where levels are reassessed and promotions/raise are given where appropriate. So that's a path that can be taken.