Anonymity is one of the desirable requirements of voting process in elections. Are there states where voting is not anonymous i.e. who voted whom is public?
It seems that Nigeria uses an open ballot system in order to minimize potential for election fraud.
As @CortAmmon states in the comments above, public voting is desirable in some cases. For example, in a representative democracy, it is important for constituents to be able to audit the votes of their representatives, to make sure the representative is faithfully representing the constituents' interests.
The Swiss Cantons of Apenzell and Glaris have a system where people vote in public, in the open air in front of everyone. The german name for that is Landsgemeinde, which translates roughtly in "Land's community".