**Short answer**
As far as I know, red blood cells and blood platelets are the only _human_ cells in our body without a nucleus.
**Background**
Erythrocytes and thrombocytes are the **only _human_ cells** without a nucleus, as far as I know. However, if you count the gut as being part of the human body (in essence it is a continuation of the skin and as such it can be considered to be on our _outside_ ), then we are _loaded_ with cells lacking a nucleus, namely all the bacteria that live in our intestines such as _E. coli_. Bacteria, being prokaryotes, lack a nucleus. In fact, there are ten times more bacteria than human cells in our gut (Wenner, 2007).
**Reference**
Wenner, _Sci Am_ 2007