I've noticed that the price of slots increases as the game goes on. Is this because of my pile of money, my character level, or my game progress? How much does it scale or increment for each bump?
The Price and Maximum Item Level you can get from the Slot Machine are based on the game host's Story Progress and Current Level.
Story Progress will generally increase the maximum level of item by a few levels at a time, and you cannot get items that are a higher level than the game host's current level.
For example, at first I thought it was based on level only since I noticed that every time I leveled up, the slot prices would get higher, however this weekend I noticed I wasn't getting any items higher than lvl 18 even though I was level 26. I did one story mission and it suddenly jumped to lvl 21 items, and one or two more story quests after that, and it was giving me lvl 26 items.
I'm sure there's a list somewhere online because my boyfriend knew how many quests we needed to do to get items our level, but I'm not sure where that list is. If I find it I'll update this answer.
As per this article the slot machines at Moxxxi's place will get more expensive as you progress through the game.
It is worth noting that some of the items you can win by playing the slots scale with your character - this means that playing the slots at a lower level will cost less while still potentially providing benefit to you at the later levels.