Seeing as I've only been doing this a year, I can't speak from the experience that some have... but I don't drink a lot of beer, so I get to see how age affects things. I can speak to three batches with regards to age.
Batch #1: Brown ale (extract/specialty grains), about 5.5% ABV. Fermented a bit too warm, was still a good beer, got really good at four months. Just had a bottle last night, one of my last few - age is now ten months in bottles (plus another month split between primary and secondary). I don't detect any loss at all from that change it went throgh at about four months. I'll be sad to see this one all begone.
Batch #2: Leffe Blonde clone (partial mash), about 8.25% ABV. Did a four week primary, no secondary. Took six weeks in bottles to get really good, but three months or so made it a great beer. Today, this beer is seven months old (in bottles). The light hop aroma it once had is gone, but the beer is as complex and flavorful as it ever was.
Batch #3: Imperial nut brown ale (extract/specialty grains), 8.66% ABV. Did four weeks primary, three + months secondary. Has been in bottles six months now. Opened the first bottle at eight months post brew (five months in bottles). Good beer, but needed more age.
A month later, beer was better. Recipe said that it would be good at six months, great at nine, amazing at a year. I plan to continue aging it and see how it goes.