IJN cruiser Kumano from the Avalon Hill game Midway. 1969 counter on the right, a somewhat newer one (ca. 1980) on the left
Comments on mounted vs. unmounted maps below prompted me to consider the durability of other common game components. Maps, as a matter of fact, even the paper kind, are rarely the component that fails first in my experience.
Usually the first component that starts to disintegrate is the rule book, especially if it's a complex game or one with poorly drafted rules and you're in the darn thing all the time. Good quality paper and printing is really useful in rule books, but this is an area where many companies scrimp an use cheap paper. Still, there's not an awful lot that can be done about this, past using good quality paper and writing clear rules. A rule book is necessarily going to get a lot of handling by sweaty fingers.
Another component that should use high quality material (but often doesn't) is cards, if the game uses them. It's my practice to buy extra cards if the game uses them a lot and it looks like I'll get to play it a lot. I bought an extra set of cards for Up Front, for example, although I still haven't busted the seal. While my original card are still usable, they are showing some wear and given the likelihood that Up Front will never be reprinted I'm glad I have it. Some other games where I bought back up card sets included Commands & Colors Ancients, Gunslinger and Memoir '44.
Finally there are the unit pieces. The winner in the durability contest are figures hands down. While occasionally they can get damaged or destroyed if stepped on, for the most part they remain in near pristine condition if minimally cared for. I have copies of Broadside and Dogfight from the 60s that I bought on eBay that are in pretty tough shape in many ways, but the planes and ships are in good shape. I have metal painted miniatures that are decades old and I'm quite certain will be around long after I'm gone. Even the pre-painted miniatures from collectible games I expect are virtually immortal so long as they are not abused.
Nearly as good are wooden blocks. I have had some cases where stickers have come off a block or two, but with the exception of an old copy of Rommel in the Desert, in no cases has it been common and many of my block games are getting long in the tooth. And those block games that have embossed or screen printed blocks seem as permanent as the plastic figures. They could be used as grave goods.
Card board counters, on the other hand, are another story. The oils from hands and the friction from the playing surface and stacking means that counters can get some serious wear, even if the game isn't played often. Back when Avalon Hill was in business I ordered replacement counters for a number of games over the years, such as Afrika Korps and Midway. Most companies don't offer this service though, and I can see problems down the line for some games in my collection. The main reason why this isn't a bigger problem is that I have so many games that few get the kind of intense play that AH games got back in the day. And many popular games go though multiple printings, sometimes with multiple companies, so that you can get a newer (Perhaps improved) copy later. Some games I have in my collection that are NOT the original copy I owned include OGRE, The Awful Green Things From Outer Space, A House Divided, For The People, Bitter Woods, Monopoly, War of 1812, Wizard Kings and Quebec 1759.
Another factor with cardboard counters is different degrees of wear. It's often the case that some counters and markers see more use than others. It's not unusual for the turn marker to wear down to a nub first. In games where you have to draw activation chits these often wear out fast as well. Sometimes a unit spends a lot more time on the board than others. The German panzer regiments in Afrika Korps often show a lot more wear than any other pieces -- while the British substitute units are sometimes in mint condition! Usually this doesn't have an impact in play, but I could see times when it might matter, such as if the game uses inverted units.
Counters do have many advantages, of course. They can hold much more game information than blocks or figures. They cost less to make and weigh less when shipping. You can stack them. So lots of times they're a good choice. But there is a trade off in durability.
One thing that game makers could do more often is offer replacement parts. I think cards should always be available for separate purchase. You shouldn't have to buy a whole new game just because the cards wore out. Rule books should be online. That way they can be updated easily and players can download a new and updated version oft he rules when their original comes apart. And I think counter sheets should be available. One thing I also hate seeing is blank counters on a counter sheet. Put something on it. Extra "Game Turn" makers never hurt. There's no excuse for blanks in any game that uses any kind of status markers, either.
For me, because I tend to dabble in a large range of games rather than play any one game intensively,. durability usually revolves around storage issues. But a lot of people like to play the heck out of their favorite games -- and for that sort of player how well the game components stand up to repeated usage is important. It's a real shame to have your game fall apart before you're tired of it.