Now I know that there are limits technically on size of plants as seen in this post, but what I want to know is what limits a plants speed? I'm guessing it has to do with the soil, but is that all? The reason for this is to create a jungle or forest that regrows within minutes of utter destruction in a wide area from a orbital barrage. There would be some left like bits and pieces, but otherwise mostly destroyed. Is that possible or just a magic handwave moment?
Well Bamboo is the fastest growing plant on Earth and it can grow (under ideal circumstances) up to 3 feet in a single day. The gardens in Alaska do so well because in the summer months they have 18+ hours of growing light available every day. The plants almost don't stop for 3 months!
Some of the things that go into plant growth are sunlight, soil and water. Any plant needs at least these three things. The more sunlight and water available the more growing time a plant has. Soil providing what the plant needs is also very important.
Now if bamboo can grow 3 feet in a day, and most of it grows in the tropics, so we'll say 12 hours of daylight growing time. that would be a foot for every 4 hours or 3 inches an hour.
If the plants/trees have have a massive root system that stores the needed nutrients to grow when the plant body is destroyed, then you might be able to double that growth rate for plants evolved to such an environment. However, I expect there to be a physical limit to how fast cells can divide and grow. Since you still need to build the structure to support the plant as it gets bigger.
Limitations to plant growth are construction materials, growth conditions and available energy. If a plant has the capacity to grow 3 feet a day but the temperatures are sub-optimal and sunlight too weak, then growth won't near the maximum possible.
Compare the growth of a plant to building a house. If the construction materials for the house aren't available then construction will slow down. If the workers are always tired because of lack of food then construction will slow. If it's too hot or too cold then the workers will have to spend more of their time cooling down or warming up thus slowing construction speed.
To repopulate a devastated area has to deal with a couple of problems. If the devastation has sterilized the soil then the soil bacteria that facilitate growth will need to be put back. If the seeds for the plants have been killed then they will need to be redistributed. If the devastation has evaporated much or all of the water in the soil then water will need to be reintroduced. Some of these steps can be done concurrently but not in the time scale of minutes. That's magic.
There are three inputs to the rate at which a plant can grow. Nutrients from the ground (including water), carbon dioxide from air, energy in the form of sunlight.
A plant will try to keep these in balance. If it's shaded it will grow more leaves than it would need just to absorb CO2 (and of course, it will grow towards the light). If it's got all it needs above ground it will expand its root system to fetch more nutrients. Both assume it hasn't yet reached its mature size. Sometimes the lack of a particular trace nutrient in tiny quantities will stunt a whole plant.
Bamboo is a bit of a "cheat". It has prefabricated a large bud (a set of concentric cylinders) underground, in the months before it starts shooting up. It can expand upwards very fast by growing in much the same way that a portable radio's antenna "grows" when you pull it out. Other fast-growing plants are often moving pre-stored nutrients from below ground (bulbs, corms, tubers) to make new tissue above ground. Spring bulbs make a dash for it in early spring, grab sunshine, flower, set seed, and store all energy back into the underground bulb when the trees come into leaf and shade them.
I wouldn't rule out something alien sprouting back in hours, if not minutes, provided it wasn't able to do that twice in rapid succession. What evolutionary pressure would cause this to evolve? Perhaps rather frequent forest fires (high oxygen atmosphere as well as high in CO2), and aggressive shading-out/ parasitism by other plants? I think your alien forest would be like a hybrid of the interesting features of bamboo and Japanese knotweed (and maybe mistletoe, venus flytrap, sea-anemone, yes that last one is an animal), plus a photosynthetic mechanism more efficient than that evolved on Earth, and maybe a hotter/brighter (blue) sun to power it all. Does it also eat insects? Small animals? People?
A vaguely remembered sentence that I can't quite place: (name) disturbed a (name of large armoured herbivore), which scared him, so he climbed the nearest tree, which ate him".
There are some great answers here based on botanical and biological reasoning. I'd like to advance an even more basic physical consideration. Plants are made of matter and matter (well, matter plus energy) is conserved. So, how much matter constitutes a forest? Quite a lot. If that's supposed to spring back up in a matters of minutes, where is all that matter coming from? Yanking it straight out of the earth underneath sounds like you're gonna get a sinkhole. Pulling it out of the air will create a massive vacuum. There just isn't enough heat and light around to convert directly to matter since matter is so energy dense (the $c^2$ factor in $E=mc^2$ assures that). So it looks like a real problem in imagining conventional plants behaving this way.
All that said, rather than raining on your parade, maybe you can roll with it and make some unconventional plants that circumnavigate the issue. Assuming that the forest does come back extremely fast, it can't have a lot of matter in it or we'd run into the problems above. So it must be of some sort of hyper-low density material. I'm envisaging plants made of a very airy, gossamer-like material. The plants could still be sufficiently tough to impede navigation (maybe they are poofballs of some carbon-nanotube kind of fibers) and opaque enough to severely limit visibility.
My understanding of terrestrial plant growth is that it is rarely limited (in the average case) by sunlight. Typically it is limited by availability of "biologically available nitrogen". After that it is probably limited by water or non-nitrogen nutrients (e.g. sulfur).
Of course in some case the ranks can be reversed (e.g. in a desert water may be the limiting factor).
If all required nutrients and H2O are over-supplied, then the limit to plant growth is the efficiency of photosynthesis in terms of the gain in energy from converting H2O and CO2 to carbohydrates and oxygen. Terrestrial photosynthesis requires light photons of the correct wavelength.
It may be possibly to bio-engineer plants which can utilise more of the full spectrum of the suns light, but this would only help boost growth if natural sunlight is the first limiting factor.
Likewise, it may be possibly to bio-engineer plants to use alternative photosynthesis mechanisms which are more efficient for a given light spectrum. This could be useful for alien plants or for plants grown in artificial space habs, but I doubt you could much improve upon existing terrestrial photosynthesis for growth in standard earth light.