It started off innocently enough, that appealing little 4 inch plant there in the nursery just begging you to bring it home.  You may even have been delighted in the beginning as you watched the thing thrive.  And thrive. And thrive. Now, though, it has gobbled up half your border and looks longingly at your neighbor's yard. Let's face it -- you brought home a thug. If you had only known.....
  Yes, plants can become a pest. Some reseed where you don't want them.  Others colonize by underground runners or creep above ground, rooting as they go. In many cases, they fulfil the very purpose for which you purchased them, but just a bit too well -- plants sold as groundcovers, especially,  might come with a caveat that you may be in store for a battle some day.

I've compiled a list here of some known suspects --  the  thugs,  bullies, and wandering derelicts of the horticultural world. It isn't intended to be absolutely comprehensive, as it's tough to keep up on *all* the plant criminals. Also, many plants are a pest in one area, but relatively benign in others, so your climate plays a major role.  Still, it's always a good idea to treat with caution any of those plants with a questionable reputation.
 

Achillea -- aka yarrow, some are much more invasive than others.  A. millefolium and A. ptarmica are the worst offenders.

Aegopodium -- usually sold in the variegated version, it will spread especially vigorously in the shade.

Ajuga -- some forms much more agressive than others.

Anemone -- A. Sylvestris,  A. hupehensis/hybridus and a few other lesser known can spread far and wide.

Aquilegia -- Perhaps more of a thugette than a thug, columbines are often less than shy in their reseeding.

Bulbs -- scillas, muscari, alliums,  crocosmia, and several other bulbs will naturalize freely, and may overwhelm in time.

Campanula/adenophora -- most pose no problem, but a few spread vigorously through underground roots and some reseed copiously.

Cerastium -- not difficult to control, but capable of invading the crowns of other perennials.

Convallaria -- lily of the valley can start off slowly, but it can sure be a determenied thing.

Cymballaria -- I've known this one to thrive clinging to the side of an old barn.  It's really happy in the ground.

Digitalis -- Common foxgloves have colonized western Oregon, and can venture where they are not wanted in other areas.

Duchnesia/fragaria -- duchnesia looks like an ugly strawberry, and makes up for its lack of appeal in zealousness.  Some of the ornamental strawberries can be a nuissance, too.

Erigeron -- most erigerons are just fine, but E. karvinsyanus can get a little out of hand.

Escholtzia/papaver -- California and other poppies can reseed all too freely in many areas.

Euphorbia -- Quite a few will seed about and several spread too quickly to come without warning.

Genista -- come to the Northwest, and I'll show you what broom can do.

Glechoma -- ground ivy is often sold as a variegated basket stuffer.  It then often proceeds to turn green and stuff your lawn.

Grasses/bamboo -- some have been outlawed in certain states. Others are as benign as can be.  There are spreaders and reseeders both, so do some research before planting.

Houttuynia -- no matter how you pronounce it, this dogwood relative has a large appetite for space.

Hypericum -- H. calycinum is the best known of the genus. A shame, since it's the bad apple in an otherwise very nice barrel.

Lamiastrum -- lamiums are borderline thugs, while this close relative is less well bred.

Laurentia -- blue star creeper is one of the more deceptive thugs.  Don't let it fool you.

Lysimachia -- a diverse genus does include some that play nice.  Others gobble up ground with abandon.

Linaria -- both annual and perennial forms will seed about.

Lychnis -- the common rose campions, especially, are quite the fecund lot.

Lythrum -- the subject has touched off many a spirited internet debate.
Suffice to say, all forms are illegal in many a state.

Malva -- I've has M. sylvestris seed into cracks in the concrete and thrive there without water.  Need I say more?

Matricaria/Tanacetum -- Little daisies by the thousands. Make that millions.

Meconopsis -- "What!", you say? Though the blue fellows are notorious for their exacting demands, the welsh poppy, C. cambrica, seeds about like there's no tomorrow in some climes.  My pretty pest.

Mints -- well, they smell nice.

Myosotis -- nope -- it won't let you forget it.

Oenothera -- Some evening primrose are perfectly safe. Others, like O.
tetragona, spread quite vigorously. At least one, O. speciosa, acts like a
rampaging hoard.

Pachysandra -- it just keeps on going, and going.....

Physalis -- The Chinese can have their lanterns back, thank you.

Polygonum -- Run away from most of these guys. Fast.

Solerolia -- now, why do you think babies cried tears over this one?

vines --  I do believe there is an entire subculture in the South devoted
to recanting tales of terror about Kudzu. Here in the northwest, english
ivy has invaded many a forest.  With any vine, just make sure you know what you are getting in to.

Vinca -- o.k., so how many of you *haven't* seen what this one can do?

Viola -- seems different violets plague folks in different areas.  Well, if
plague you can call it.