Re: what are the types of cells that do not reproduce?

There are a couple of concepts which are useful to review in light of this question. First of all, 
cells can differentiate into other cell types without undergoing "reproduction", or cell division 
(also termed "mitosis", or "entering cell cycle"). For example, undifferentiated stem cells can 
become blood or liver cells through remodeling their components and changing their gene 
expression, without having to divide. There are other cells which will never change their type 
(terminally differentiated), most of which can still divide and some of which can not, also called 
"replicative senescence" or "quiescence". Importantly, these limitations to division can be 
overcome in the case of cancers. 

You are right in that red blood cells (erythrocytes) lack a nucleus and do not divide. Gametes 
(sperm and ova) also cannot themselves undergo further division after they arise through 
meiosis. Muscle fibers also do not divide- these multinucleated "cells" arise through the fusion of 
hundreds to thousands of mononucleated muscle cells. As a final example it is generally thought 
that adult neurons cannot divide; although some have been recently found to express cell cycle 
proteins there is still argument over whether these neurons can produce new cells or simply 
expire through programmed cell death (apoptosis). Neuronal replacement can nonetheless occur 
slowly through differentiation of stem cells. These issues are medically relevant especially in the 
case of Alzheimer's disease, in which misfolded amyloid proteins can push neurons down this 
abortive division/death pathway.