Carolyn, TAfrikanski may not know it, but I think he's right on the money (tee-hee) with the pizza slicing. A couple of years ago I researched this for one of my American Culture classes. Well, come to think of it, you probably know how the quarters came to be, but for anyone else who wonders why Americans have them and we don't, I could shed some light (and you can correct me if I mess up your history).
So the 1/4 denomination wasn't unique for the US back then and neither was the word "dollar". Charles II of England struck coins for Scotland - a dollar, 1/2, 1/4, 1/8 and 1/16. The United States began cutting their own coins in 1794. But between 1776 and 1794 the Royal Mint didn't provide (go figure) the coins needed for commerce, so the Thirteen Colonies used whatever foreign coins they had available. Like the Spanish dollar. The Spanish cut the silver they mined into coins worth eight reales. Most of those were shipped to Spain, but some were used locally. They became very popular in the colonies and were used well into the 19th century. The Charles II dollar fractions were used, but Spanish coins of lesser value were scarce. So what did the people do? They cut the silver Spanish dollars into pie-shaped halves, quarters, eighths and sixteenths - the famous "pieces of eight" in romantic literature. The very first quarters that were cut weren't even labeled as 25 cents - that came later. New denomination was accepted, like the 10c, but the quarter remained popular to this day.
Why don't we have quarters? Well, I haven't done research on Bulgarian coins, so this is just what I know from history books.
After our liberation in 1878, there were all sorts of coins that were left in the newly freed Bulgaria - German, Austrian, Hungarian, Russian, Dutch etc.. Some no longer used in their respective countries. A lot were so worn out, people told them apart by size and shape. So we needed coins. The National Bank was promptly founded in 1879 and a currency was chosen on June 4th 1880 - Lev, made up of 100 stotinkas. "Лев" comes from an old form of "лъв" which means "lion" and "стотинка" translates into "little 1/100". The first copper coins were cut in England and were 2, 5 and 10 stotinkas, as was decided - no 1 stotinka coin. Later came the silver 0.50 and 1 and 2 levs, made in Russia. The silver 5 and the golden 10 and 20 levs were struck several decades later. The myriad of foreign coins in circulation is gradually replaced by our own. And the 1-2-5 tradition has been followed ever since. I don't know how the decision was made, probably this division was the most popular. So we never had and were never supposed to have quarters. But here's the funny thing: Before the introduction of the lev and its rates to existing coins, all financial accounts (and the country's budget) were kept in franks. We loved the French. Then there was also the whole "Russian occupational government" thing. So we had Russians governing us. (This I researched now) Both the Russians and the French had coins of 0.25 yet we managed not to copy either system.
Well… I'm sorry, Carolyn. We were so close to a quarter, but alas… :P
I hope this rather long post wasn’t too boring. Apologies to anyone who fell asleep reading it. :)
PS: Some of the theories about the origin of the dollar sign are connected to the Spanish dollar - a slashed eight, the superimposed PTSI (these coins were cut in Potosi) or the pillars with ribbons around them, depicted on said coins.

Man, I need some synonyms for "coin"...