About as it does today. Change below a quarter is so valueless that it's very difficult to use. There aren't any products of value you can buy for a penny. To convert change into "real money" you usually need to go to a change counting machine (like a coinstar) or roll up your coinage and take it to a bank.

Let's bring in some math. Let's say every day one engages in 2 financial transactions, and let's say at each one loses out half the value of $.25 due to rounding. That's $91 a year, which is less than 2% of the per capita GDP of folks in 1909. That's a reasonable upper bound to how much money would be "left on the table" due to the smallest value of coinage. In practice it would be far less because people would just refuse transactions if they thought they weren't getting their money's worth. The simple solution to an item not being valuable enough to be a good trade for the smallest denomination of coinage is to ask for more than one or to have it added for free to a more substantial transaction.

Also remember that the economy was very different in 1909, people weren't shopping at stores multiple times every day (starbucks, sandwich shop, etc, etc.) and there were fewer mass production goods available.

today we could stop circulating pennies, nickels, and dimes.

Yes, we could. Pennies are already beyond obsolete and will probably stop being produced in the near future. It costs almost.02 to produce a penny. Canada is eliminating the pennny: http://www.nytimes.com/2012/03/30/business/global/in-canada-the-pennys-time-to-shine-is-over.html

Rising metal costs are contributing as much to the stupidity of small coins as inflation is. Copper has increased in price massively, like 500% in a few years. Inflation has made a handful of pennies worthless; we continue to give people "proper change" because our culture is obsessed with perfect accounting when it comes to money. If we are going to track pennies you may as well calculate to 8 decimal places when you accrue interest...

If you are interested in reading about a time when proper coinage was really a problem, do some research on early Greek coins. Before the Greeks struck some large silver veins, they were stuck with gold coins or (at best) electrum (a mix of gold and silver of unreliable proportions). The smallest gold coin might have been worth more than an entire person's personal wealth. Thus paying in gold coins was only really viable for huge purchases, like a city buying a big shipment of grain from another city.

But once you get coins of a reasonable value, you really don't need extremely small and specific denominations. You can adjust or haggle over the amount of product as a way to correct the difference in price if it is between denominations. More importantly, the whole concept of highly exact price figures and things like "Only $29.99!!!" is a byproduct of consumerism and marketing which is a relatively recent phenomenon in the last century and a half.