I am due with my second child in Sept., and my son will be 19 months old at that point. I was forced to stop breast-feeding my son when he was 6-months old because of supply issues (I had supply issues his entire life, but the milk was finally tapped out at 6-months). I fully intend to reintroduce breast-milk into his diet if I am able to produce enough with this second child (no guarantees that I'll be able to produce enough this time around, but I have hope) for the health benefits provided to him.
In some Asian and African countries, breast-milk is considered advantageous into adulthood. Not that adults feed at the breast, but in Mongolia, women leave their excess breast-milk for their husbands and older children to drink as a treat, and it is a standard compliment to say that a man must have breastfed into his teens (as a comment on his health and strength). As a note, they do not breastfeed teenagers in Mongolia - it is ONLY an expression. Although a child at 18 months should not be living on breast-milk alone, I believe there are still physical advantages.
However, I do not intend to reintroduce breast-feeding because my breasts are no longer a food-source to my son - they are a body part. This is not because of his age but rather because of the long period of time that has passed since my breasts were a food-source in his mind, and I doubt he remembers how to do it. I know that, when I hold him in the shower and whatever, he makes no attempt to nurse. I intend to give him (if I am able) breast milk in a sippy-cup or glass to drink, which I think will diminish any psychological problems that even Freud could predict.