I had to laugh when I read that Plectranthus Amboinicus is an invasive plant. True, it is extremely easy to propagate my trademarked semi-succulent Plectranthus 'Velveteen Aromatic'™ and I currently have 22 copies of it. All of them are in pots. When I lived in South Dakota (Zone 5, same as my current location in Montana), I planted it in the ground a few times, and sometimes it thrived through early autumn. But it did not take over my yard because all you have to do to control it in Zone 5 is wait for the temperature to hit 32 degrees and it turns into a ball of mush. This is my most bushy 'Velveteen' shown with a (somewhat) related Coleus.
Reading more details, the plant is considered invasive in tropical places. Many invaders that thrive in warm areas are not a problem in Zone 5. Kudzu may be the most famous invasive plant in the USA and is called "The plant that ate the South." For fun, I looked up its growing zones, and it shows as zones 7-10. The Nature Conservancy claims Kudzu can survive Zone 5, but struggles to establish deep mats due to the cold. I looked up "Kudzu Montana" and found that the botanical name is Pueraria montana. Other than the name, I couldn't find any links to Montana. There are some advantages to living in cold places. Another is that there are no pesky insects for half the year.
Other examples of plants that are invasive in warmer areas but are controlled by the cold in Zone 5 are Butterfly Bush, Lantana, Japanese Honeysuckle, and Sweet Autumn Clematis. The first two have crossed my radar as possible acquisitions, but I currently have no plans for them. There also are invasive plants that have close relatives which are not considered invasive. Trumpet Honeysuckle is not as aggressive as the Japanese variety. Other than the Oxeye daisies in a pot on my back patio, I'll just avoid the invaders. If anyone thinks this single pot of daisies is dangerous, we have a neighbor who encourages these plants instead of having a lawn. I'll deadhead my pot to keep it from going to seed; I doubt that my neighbor will do the same with his thousands of blooms.
What about invasive succulents? Creeping Sedum is the only example I could find of a succulent that was hardy enough and aggressive enough to be potentially invasive. I have three varieties of SunSparkler sedums that are cold hardy but supposedly are not invasive. The other cold-hardy succulents I have are Sempervivums and Delospermas, and both grow too slowly to be invasive. In fact, three months after acquiring two of the Delosperma 'Fire Spinner' from The Three Company via Amazon, sometimes I wonder if they are alive. They are green but have shown no sign of producing the vivid flowers in the photographs.