Qyburn explains the significance of the Maester's Chain, a symbolic ornament earned by students of The Citadel, with each link representing the mastery of a different subject.
Qyburn:You can hear a maester coming long before you see him, thanks to the jangling chains he is forced to wear around his neck. It's meant to remind him of his servitude, like the collar of a dog. He can't remove it, even to sleep. Each link signifies mastery, or at least what those withered old men in The Citadel consider mastery, of a different area of study: gold for the study of sums and accounts, silver for healing, iron for war, black iron for ravenry, lead for poisons and Valyrian steel for the... higher mysteries. Only one maester in a hundred forges such a link, only one maester in a hundred has a sense the gods gave a goat.
Though I admit that no one had performed true magic in centuries. We sit in a room mumbling over a candle of dragonglass, trying to make it catch fire. After a sleepless night of failure, we are supposed to admit to our own limitations. To win the link, we are supposed to lose their curiosity. I never did. In time, the archmaesters decided more links were useless to a man who wouldn't be chained by them. They took my chain and expelled me from the Citadel. Their loss.
The chain as a whole is supposed to signify the realm. One cannot have only lords or only knights, one needs farmers, smiths, merchants, shepherds and the like. Like a chain forged of many different metals. An obvious and trivial point, disguised with pomposity, much like the maesters themselves. They study without learning and proudly pass down the same knowledge that was passed down to them, with no addition. Perhaps such is to be expected when one considers the kinds of men who become maesters. The youngest sons of noble families, dutiful and timid, raised in the shadow of their older brothers; or bastards and peasant boys whose minds are easily satisfied by the knowledge of their next meal.
Because bold men will not be chained. They dare to ask questions that the maesters fear to answer. They will look at a living man and ask: "How?" ... They will look at the dead and ask: "What if?".