Homo sapiens is, as far as we know, the only species that has permanently enlarged breasts. In the rest of the primates, plump breasts last only as long as breastfeeding does:
Humans are the only primate to have breasts that are enlarged at all times, in contrast to other primates which have swollen breasts only when breastfeeding. This suggests that human breasts have a unique function that is not shared by other primates. (source)
The main reason for larger breasts existence (which would explain their evolutionary presence) seems to be not lactation -pre-pregnancy size is no indication of a woman's ability to produce milk-, but because they are a signal of a woman's ability to store fat - and therefore be considered more fertile (Mascia-Lees et al., 1986, Pawlowski, 1999, Pawlowski et al., 2004).
Women with hourglass figures have been proven to be more likely to become pregnant:
Women with higher breast to under-breast ratios (large breasts) or low waist to hip ratio (WHR) have higher hormone levels (...) High levels of these hormones are good indicators that a woman will successfully become pregnant.
The ratio of bust-to-waist may predict hormone levels, so women with larger breasts are thought to be more fertile, and more attractive. This is believed to be the main reason why humans have permanently enlarged breasts.