Probably one of the large invasive knotweed species such as Japanese knotweed or its hybrid Bohemian knotweed (which is often misidentified as Japanese knotweed - and is actually more common than the Japanese parent in some regions such as mine for instance).

fallopia shoots - Bing images

Carol, that was my first thought as well :-) Can't be any kind of invasive knotweed if it only grows an inch or two tall. However, previous encounters with Houttuynia indicate it usually produce much more dense shooting of new growth than every 12 inches or so!

@lpindmv, You can keep the unwanted invader from encroaching into your planter if you install some sort of solid barrier at the base of the fence....like any kind of sheet metal. You will still need to manually remove what has already crept over.

That one doesn't produce a tiny top either. But I suspect the point of

It does not get much taller than this. I try to get them while they’re babies, but I think the largest I’ve seen is around 3” across

was the growths on this side of the fence are all being removed before they get very far along.

Thanks, everyone. Here’s a photo of my neighbor’s planting area (over the fence, so pardon the poor quality — and sorry for the invasion, neighbors!). As you can see, it does get denser, but doesn’t get taller. I’m not familiar with Houttuynia cordata, but from Google, it does look close!

It is absolutely Houttuynia cordata!!

And has also just come out of the ground on his side, will not be staying that short unless he is mowing it in order to produce a sort of turf and stones effect. Or some other dwarfing circumstance is present.

Thank you both. Reading up on this plant is not reassuring! I’ll approach my neighbor about trying to remove it on their side and plant something less invasive like hostas.

There is an implication that there is something haywire with the root barrier installation which should have taken care of the problem.

Yes. It’s helped, but not eliminated the invasion. There’s a mature ornamental tree a few feet from the fence line, and several thick roots about 8” deep ran perpendicular to the fence. I of course didn’t want to cut them to get the barrier in deeper, but now I fear the rhizomes are going under!

ive seen it pop up and the other side of a cement sidewalk ... you arent going to barrier it into submission ...

Houttuynia has certain proclivities to world domination ... and i spent a decade trying to kill it with round up at the parents old house ... i lived an hour away at that point.. so i cant say i did a thorough job of trying.. lol ... i eventually convinced her to sell the house.. and leave it there.. lol .. of course she was 85 by then.. so really.... there were other reasons .. lol ..

digging it up alone wont work ... you dig .. and then 6 weeks later.. spray what pops back up .. and then 6 weeks later.. spray what pops back up ... for what will seem like forever ...

good luck.. you will need it ...

ken

btw .. this has been discussed soooo many times .. just so you dont feel alone out there see what other think or have done :

https://www.houzz.com/discussions/perennials/query/Houttuynia/nqrw

https://www.houzz.com/discussions/1700184/what-were-your-worst-invasive-experiences#n=33

A cement sidewalk is not a vertical barrier - I've had Rosa rugosa 'Alba' i was growing come up across the street.

And yes this was definitely sucker growth and not reseeding.

Running bamboos are another type that may pop up yards away from the source plant in one go. So that I've had 'Koi' golden bamboo travel at least 15 ft. before emerging.

It has been stated that honey fungus (not a plant of course, but still something that spreads through the soil in a root like fashion) can creep 60-100 ft. in order to find additional plant material to infest.

But nothing traveling through the soil can get past an impenetrable vertical barrier without going over or under it.