The answer really depends on how you think of invasive. One extreme answer is to say that all things are relative, and that the concepts of local and invasive are all relative. This matters to a certain extent because we (ecologists) draw a fuzzy line between invasive and naturalized (http://onlinelibrary.wiley.com/doi/10.1046/j.1472-4642.2000.00083.x/abstract). You could start with some basic species that we all think of as either good, local, or neutral. Take the earthworm. Most people think of it as a common native species, but it's actually an invasive species that has radically changed much of North America that came over with the Europeans (http://www.dnr.state.mn.us/invasives/terrestrialanimals/earthworms/index.html). Similarly, brown trout are also invasive, coming to the US in the 1800's (http://www.dfg.ca.gov/fish/resources/WildTrout/WT_BrownDesc.asp).
As far as why invasive species succeed, it's still an open question. What you reference is the enemy release hypothesis, but there are others such as disturbance, diversity of the new community, and just the traits. Here's a good review: http://onlinelibrary.wiley.com/doi/10.1002/ece3.431/abstract http://onlinelibrary.wiley.com/doi/10.1111/j.1461-0248.2009.01418.x/full
To answer your question about adapting, I wouldn't trust your memory. 759
It's important to keep in mind that most likely your memory is flawed, and you are only observing a very limited number samples with an incomplete sampling methodology. For instance gypsy moths undergo wild population fluctuations, and have for over 100 years. (http://link.springer.com/article/10.1007/PL00012004). How well do ecosystems bounce back from invasions depends on lots of factors. Often time the ecosystem can survive, but the some species won't. An easy example would be Norther Hardwood forests. Chestnuts (extirpated by an invasive fungus)are mostly gone from the wild, but the forest persists, as well as many of the species that rely on them. So in part the answer to your question depends on scale. The forest survived, but an individual species was nearly destroyed.
Ecological economists have tried to put a value on the destruction that invasive species have caused too (http://www.sciencedirect.com/science/article/pii/S0921800904003027). 3