IPAs go bad, barley wines improve, and stouts mellow over time. I'm curious though: what's actually happening? What compounds are building up or breaking down? What chemical processes actually happen to constitute "aging" in beer?
In short, the main factors in aging is a combination of compounds breaking down and oxidation.
IPAs - hop aroma breakdown
For IPAs, it's the hop aromas that break down fairly quickly - the aromatic compounds are volatile and break down noticably within a short timeframe:
Barley Wines - Oxidation
Barley wines are not hop flavor or aroma focused beers - the major hop component is bittering acids, so they don't suffer the same problem with aroma losses as IPAs. However, some of the bittering compounds also have a half-life of 1 year, so clear changes are happening in that timeframe to the bitterness.
However, the major change for a barleywine is oxidation. There is always oxygen in the beer - either small amounts introduced during packaging, or from oxidization during the mash, which is then later released into the beer. When alcohol in the beer oxidizes, it becomes ethanal (acetaldehyde) which is a main component in sherry, and gives apple or sherry-like tones, which in a big barleywine are considered positive effects of aging, if kept in check.
Contrasting with both IPAs and barleywines, Stouts that are meant for aging are neither hop focused nor benefit much from oxidation. Oxidizing the medium roasted malts in the wort would produce cardboard flavors due to production of trans-2-nonenal. Fortunately, the highly kilned malts act as an anti-oxidant to help prevent the beer from oxidizing. What little odization does occur tends to produce port-like flavors.
The roasty character in stout comes from large Maillard compounds. These are still poorly understood, I can't find research to back it up, but I would guess the roasty character mellows due to the large compounds breaking down into shorter ones. This might explain why in some stouts the dark fruit flavors become more emphasized over time (since they are shorter Maillard compounds.)