Have you ever heard "Running is so cheap! The only expense is buying running shoes!"? It's true, if you run nude. If you run farther than a tenth of a mile outside, you will probably want to buy some shorts, and not cheap ones from Target because there are hideous seams in cheap shorts. Don't understand? Go for a run in a pair of $8 shorts and you'll see what awful things happen to your legs. You'll also need a shirt that's not made of cotton because it will rub your skin off. So, you'll need at least two of those. And if you are a female, you'll want a sports bra because even more terrible things happen if you don't wear a good one. The good news is that I've had only five shorts (all black and white Nike Tempo shorts) for the entire time I've been running. I get two new shirts a training season and usually only one pair of shoes a year.
So that's it! All you need to spend money on when you run are shoes, shorts, a shirt or two and a sports bra.
Unless you want to improve your time/distances. Then you may want a GPS watch or an app on your iPhone. So you might want to consider an arm band for your phone, because you really should always have it with you when you are running. And if you don't like carrying your license, you should get a Road ID.
Once you get better and see some improvement, you may want to enter a race. 5ks can cost anywhere from $10 to $25 and there are local ones literally every single weekend.
When the race bug bites, you may decide to do longer distance races. A half or a full marathon can range from $50 to $100 plus. You'll need a water bottle on training runs and then some fuel (Gu or whatnot), some good sport sunblock (something that won't turn white once you start sweating or get rained on). And ibuprofen. And if you can't run outside for weather reasons, you'll need to log some training miles inside on your treadmill or at a gym.
When you do all the local races, you'll probably want to travel a little bit to see new courses. Then you'll probably want to get a hotel close to the start or finish. These are somewhat pricey, but worth it.
So, yeah, all you really need to spend money on when you run are shoes, clothes, entrance fees, Gu, something to hold your phone and identification, a water bottle, hotels, and something to display your medals.
Speaking of which, I just bought new shoes. I got the same model of Asics: Gel Nimbus 14s, but a different color. I've had these things since October and my knee started to twinge which is usually the first sign I need a new pair.
Zappos had them listed for $140. The odd color was $110. I got them off Amazon for $80, free shipping. Zappos, you are not as awesome as you used to be.