There are two things to know about the guy who's spent his entire life serving the de Boys family: He's crazy old and he's crazy loyal. When Adam learns that Oliver wants to kill Orlando, he gives Orlando his entire life savings and flees with him to the Forest of Arden for safety.
This is a pretty big deal because conditions in the forest are harsh— it's icy-cold, windy, and full of dangerous animals. In short, it's no place for an old man who has never taken a wilderness survival course and who probably forgot to TiVo episodes of Man vs. Wild on the Discovery Channel.
In fact, Adam nearly dies of exhaustion and hunger in Arden. (Now that's devotion.) As Orlando tells Duke Senior, "there is an old poor man/ Who after me hath many a weary step/ Limped in pure love" (2.7.134-136). Like Kent in King Lear, Adam is a perfect model of old-school service and loyalty. He's willing to do anything for his master. (Does this make him a chump, or are his actions admirable? You decide.)
We know what you're probably thinking: If Adam is such a loyal and beloved family servant, why doesn't he show up at Orlando's marriage in the play's final act? Good question. It is rather conspicuous that Adam has gone AWOL at this point. So, where the heck is he?
Well, we're not exactly sure. Maybe he's back at Duke Senior's cave taking a nap? Seriously. Literary critic Anne Barton suggests that Adam's nowhere to be found during the play's finale because he's "simply too old to help initiate the new social order." In other words, Shakespeare doesn't want some old man (and everything he symbolizes) standing around on stage at this point because it doesn't quite jibe with the idea that the newer, younger generation (Orlando, Rosalind, and company) is now in charge. Plus, Adam might break a hip on the dance floor. (What? Shakespeare's the one who doesn't invite Adam to the party, not us.)
Check out "Symbolism, Imagery, Allegory" (specifically, the "Seven Ages of Man" speech) if you want to know more about the play's attitude toward youth and old age.