Can I start mining with $300 and grow from there?
You can literally start mining with the computer that you are using to post this topic on the forum. You just would not earn a lot since the computing power is considered nothing special (unless you have a really awesome and expensive hardware setup).
I think the question you want to ask is how much money do I need to invest into mining Bitcoin for it to become profitable by an x amount of percent, in regards to the money that I put in? That might be a bit more difficult to solve since you have to consider the costs of miners, electricity, internet, space and so on...
So the most necessary question to figure out your costs is: how much money, at minimum, do you want to make?