Be the obnoxious, needy jr programmer
One of the things that I firmly believe made my time at ReverbNation successful was that I picked up so much information about the business side of the house. It was one of the big drivers of the acquisition in our minds - get access to 2 Fuqua grads that ran the ship there and we'll pick up a ton of business and product experience during our time. This crash course wasn't thrust upon me, however, and I had to stick my neck out over the line on many occasions to push for information that wouldn't otherwise be freely offered.
To many people, I was "that guy". You know him, the guy that wants to be in every meeting and every discussion. Most of the time I wasn't interested in talking, just listening. I'd stay hours later if I had to, it didn't matter. These people had information that I didn't - information that sometimes took a whole career to learn - and I very much wanted to be opportunistic in obtaining it.
Experience is, in a sense, bound by how fast time passes. Assuming you obtain no information from your peers, there certainly is a cap on your learning potential. The other side of the coin is to be the obnoxious, needy junior developer/sales/finance/whatever guy. You're young and not getting paid near what the senior guys are getting paid, so you should seek to close that delta in other ways. Namely, your quest for knowledge.
Ask questions, lots of them. Ask for sensitive information, even if it's personal at times. Understand that you're going to cross the line - apologize accordingly. If you're worth your weight in your job role, you've earned the ears of your superiors and should use them accordingly. Talk to the CFO, the guys in support, and the other developers. Ask about fundraising and bizdev deals that you see happening behind closed doors. Auditors getting paid $400/hr for accounting stuff? Fair game if you can catch them in the break room. How did they get to where they are now? What advice to they have for young founders that will eventually need to pay for their services?
To be clear, I specifically went out of my functional area to ask questions. As a developer, I soaked in a lot of things through osmosis and working with the code base. If your company is very walled garden in terms of responsibilities you'll need to make the extra effort to get at the other departments. In the end, people like talking about their own jobs and their lives. Their successes and struggles, how awesome/terrible their customers/coworkers are.
Startups are often born out of struggles inside the building. Keep your ears peeled and your brain open.