Australia

Eliminated the one and two cent coins by 1992.

Brazil

Stopped issuing one cent coins in 2005.

Britain

Removed the half-cent coin.

Denmark

Eliminated denominations less than 50 öre coins.

Finland

The one and two cent coins are currently minted only for collectors for numismatic purposes.

Hong Kong

Stopped minting its five cent coin since 1980.

Hungary

In 2008, a ’rounding system’ was instituted to the nearest 0.05, resulting in the ending of practical use of the one and two forint coins.

Israel

Got rid of the one agora coin and five agora coin in 1991 and 2008 respectively.

Malaysia

Eliminated the 1 sen coin from its currency by instituting a ’rounding system’ to round transactions up to the nearest .05 ringgit.

Mexico

Made the five cent coin their least valued denomination.

Netherlands

Eliminated the guilder in 1980, and the one and two cent Dutch Euro coins in 2006.

New Guinea

Removed the one and two Toea coins from circulation in 2007.

New Zealand

Instituted a ’rounding system’ to round to the nearest 0.10 and removed the one and two cent coins from circulation in 1990 and the five cent coin followed in 2006.

Norway

Eliminated denominations less than 50 öre coins.

Sweden

Eliminated the one and two öre coins from circulation in 1972, the five, ten, and twenty-five öre coin by 1991, and the 50 öre coin by 2010.

Canada

Canada currently has an Alloy Recover Program where all coins that have metal value greater than face are being taken out of circulation by the Royal Canadian Mint. These coins are primarily older pennies and nickels.

United States of America

he U.S. eliminated the half-cent, two-cent, three-cent, half-dime coin (replaced by the nickel), and the twenty-cent coin. Legislation in progress with attempts to eliminate the one-cent penny.

UPDATE:

Canada eliminates their penny as of 2012! Read more: Canada Eliminates Penny
 

A LOT OF COUNTRIES HAVE GOTTEN RID OF THEIR VERSIONS OF THE PENNY, THE NICKEL, AND EVEN UP TO THE QUARTER! WHY HAS THE UNITED STATES NOT DONE THIS YET WITH THE PENNY?

The United States has had legislation pass to Congress to achieve these goals, however lobbyists have stopped the legislation from passing to date. You can read all about U.S. Penny Legislation.