Not safe enough for me to try it.
Potatoes actually contain a very dangerous toxin called solanine .
This toxin is concentrated enough in the green parts in the plant to cause solanine poisoning.
This includes the sprouts/eyes, and the potato itself if it's green.
This article from the New York Times health guide indicates that it is something to be taken seriously.
Per this article, if the sprouts have been removed, and the potato is not green then it is safe to eat as far as solanine poisoning is concerned.
However, a potato as far gone as you have described sounds disgusting .
A soft potato is on its way to going bad.
Where I am from, potatoes are cheap enough that it's just not worth the gross factor for me to eat a potato that has 10 cm sprouts and is squishy.
I do eat potatoes that have little nub sprouts on them and that are slightly less than firm, after removing the sprouts of course.
cooking.stackexchange.com/users/60/hobodave
It is safe to eat a sprouted potato if it is still firm (source: University of Illinois ); however, don't expect it to act the way an unsprouted potato would.
Part of the starch will have converted to sugar.
Be sure to store potatoes somewhere cool and dry with good air circulation.
Also, keep them away from onions.
cooking.stackexchange.com/users/364/JustRightMenus
Other online discussions suggest it's reasonably safe and the majority of us have been peeling and eating soft sprouty spuds for years.
Is this correct?
Um...
Well, I grew up eating them.
Towards the end of winter, all the potatoes looked like that.
We snapped off the sprouts, ate the firmer ones, and saved the rest for planting.
We didn't die .
I don't think.
Unless this is all a dream, the last twenty years merely the illusion of my dying, spasming, potato-poisoned brain.
That said, if you have a choice, I would stick with potatoes that haven't sprouted...
cooking.stackexchange.com/users/86/Shog9
I know not to eat a green potato, but sprouted ones are no big deal.
Just trim off the sprouts.
Knives comment was really funny...lol!
cooking.stackexchange.com/users/None/
My Dad grew up on a potato farm and he warned us about green potatoes but we were allowed to eat firm ones that had sprouted as long as we trimmed off the eyes.
I don't think storing your potatoes with herbs or essential oils would be a good idea unless you wanted your potato dishes to taste of that stuff.
I'm fascinated by the comment about not storing potatoes with onions because I always have.
cooking.stackexchange.com/users/7922/bakingnotwriting
Potatoes are simple, but 10cm is obscene, throw them out.
Potato safety 101 Don't eat the sprouts, just cut them off (they taste bad anyway).
Unless they're more than a few cm.
Don't eat potatoes that have any green tint to their skin [1].
Don't eat potatoes that are soft [2].
Tips Keep your potatoes dry
Keep your potatoes in a dark place If you store potatoes well over winter, they won't sprout and will sweeten as the starches turn to sugar and the flavor will become more complex and earthy.
Stored potatoes are delish.
Notes
[1]
This is difficult to see on purple/blue potatoes, so just eat them fresh.
[2]
You don't want your potatoes to be hard , you want them to be firm to the touch.
Don't be afraid to give them a bit of a squeeze.
As an exception, "new potatoes", which are just very young and small potatoes with thin skins, are usually a bit softer.
Source :
My friend worked on a potato farm.
Disclaimer : I am not a healthcare or food safety professional.
cooking.stackexchange.com/users/13942/dylan
The solanine is found in the green skin and also highly concentrated in the eyes, which form sprouts.
Solanine is highly toxic, but is usually found in really small quantities.
You would have to eat a lot of it to kill you.
But it's still probably not a good idea to eat in general.
Toss any crazy sprouting potatos out, learn the proper way to store them.
You wont have that problem....
cooking.stackexchange.com/users/4179/Chef
From this Smithsonian article : Not to worry though, fatal cases of solanine poisoning are very rare these days.
Most commercial varieties of potatoes are screened for solanine, but any potato will build up the toxin to dangerous levels if exposed to light or stored improperly.
Often, the highest concentrations of solanine are in the peel, just below the surface and in the sprouted “eyes”—things that are typically removed in cooking preparation
cooking.stackexchange.com/users/53139/Stacey Murphy
If you keep your potatoes in a well-ventilated, low-light place, they'll last longer before turning green and sprouting.
That said, I peel green potatoes, discard the sprouts and make mashed potatoes or scalloped potatoes out of them--because after all that, they seem kind of ugly to me.
They've never made me sick, nor anyone I've fed them to.
Maybe it IS because modern cultivars have been selected for low solanine content.
That seems logical to me.
Nevertheless, my mother did the same as I do, and I was born 20 years before the 1980's began.
Maybe the poisoning danger is at least mostly negated by peeling and cutting away any green flesh.
cooking.stackexchange.com/users/64300/Cindy Skillman
My family doctor has always advised that pregnant women should avoit eating potatoes.
As sprouting potatoes contain toxin which may cause defects in the developing baby.
Some stores break the sprouts away, but there might be still toxin left behind.
It may not hurt normal people, but for pregnant women, it is too risky.
Try notto buy potatoes that have sprouts, buy enough for what you need to avoid them sprouting while keeping them too long
cooking.stackexchange.com/users/50941/Yuet
I'm talking about potatoes that have gone somewhat soft and put out shoots about 10cm long.   Other online discussions suggest it's reasonably safe and the majority of us have been peeling and eating soft sprouty spuds for years. Is this correct?
Is it safe to eat potatoes that have sprouted?
cooking.stackexchange.com
<food-safety><potatoes>
cooking.stackexchange.com/questions/1260
cooking.stackexchange.com/users/373/Tea Drinker
Potatoes can safely be eaten if you remove the sprouts, although the taste may not be as good. Soft potatoes are turning bad and are not as safe. On the other hand, green potatoes are not safe to eat. In general, as long as the potato is still sufficiently hard, you can  still eat it as long as you have removed the sprouts and any green bits.
This is not a good idea and the potatoes are probably bad anyway. As long as you don't eat the sprouts and the potatoes are not green, then this is fine.