"lex talionis": (Latin for "law of retaliation") sometimes referred to as the "eye for an eye " principle, a retaliatory action taken with a degree equal to the original offence. Is there a word (in no particular language) that is similar to lex talionis but the retaliatory action that is taken is of a much higher degree (much more violent for example) than the original offence?
As a single word describing an "unrelenting" variety of "Law of retaliation", you could use the single word "Lawlessness". In fact, you could add the suffix -ness to several words below to add the "excessiveness" component to a retaliation. I also can't really imagine what a "Law of excess-retaliation" would be in Latin or any other language.
Although "Lex Talionis" does sound harsh, it is still based on "legal or acceptable" behavior concepts. They are described in your examples as actions of "Justice". In the case of "Lex Talionis", "actions of equal nature" results in a "just" expression of reciprocity or punishment rendered. It would seem that opposing concepts, like "excessive retaliation", could only be described using adjectives like "Unjust", "Lawless", "Ruthless", or "Merciless" as they would apply to any reciprocal action or punishment.
The US military use the term punitive measures.
The introduction gives an illustration of when they used of this measure in the past.
(in warfare) retaliation against an enemy, for injuries received, by the infliction of equal or greater injuries.
two eyes for one/an eye
Many sources explain the phrase "eye for an eye" might better be read as "only one eye for one eye". Wikipedia's article begins with this variation.
Under Babylonian and Hebrew Law, the “eye for an eye” law was intended to restrict compensation to the value of the loss; thus, it might be better read “only one eye for one eye”. Catholic Herald
An accurate reading of the biblical phrase "an eye for an eye" in Exodus and Leviticus is said to be: 'only one eye for one eye'
I believe the meaning of "two eyes for an eye" is easily understandable. Interestingly, it's also part of the lyrics in a Megadeth song:
Vengeance breeding greater vengeance, two eyes for one eye
Further, the phrase "two eyes for an eye" is much more popular (about 50,000 hits on Google search, and gives results such as these:
"Two Eyes for an Eye: The Neuroscience of Force Escalation"
Two eyes for an eye:
The role of retribution in Israel’s military thinking
Article from The Economist
I don't believe in an eye for an eye, I believe in two eyes for an eye.
Quote from Bas Rutten, famous professional fighter, link to video here
Also note that as explained in the Wikipedia article given below and the Stanford Encyclopedia of Philosophy, "retribution", contrasted with revenge or vengeance, is less loaded with connotations of personal emotions and desire to get back at someone, and has more to do with justice generally.
The idea of an "eye for an eye", or lex talionis, falls under the theory of justice called retributive justice. A word that pops up often in this subject is "proportionality". Often we see proportionality:
then thou shalt give life for life, Eye for eye, tooth for tooth, hand for hand, foot for foot, Burning for burning, wound for wound, stripe for stripe.
Exodus 20: 23-25 KJV
Sometimes we don't. The following is from the exact same chapter:
16 “Anyone who kidnaps someone is to be put to death, whether the victim has been sold or is still in the kidnapper’s possession.
17 “Anyone who curses their father or mother is to be put to death.
Exodus 20: 16-17 NIV
The reason I'm making a point about "proportionality" is that if you don't find your term and need to make one up, you'll most likely want to focus on this aspect.
Proportionality requires that the level of punishment be related to the severity of the offending behaviour. An accurate reading of the biblical phrase "an eye for an eye" in Exodus and Leviticus is said to be: 'only one eye for one eye'
Retributive justice at Wikipedia
(3) that it is morally impermissible intentionally to punish the innocent or to inflict disproportionately large punishments on wrongdoers.
Retributive justice at Stanford Encyclopedia of Philosophy
Personally I'd be surprised if there weren't a term, or at least a military doctrine or policy practised by a nation. I say this because in wartime you see this all the time, retaliatory strikes hardly ever are carried out on a measure for measure basis, they always tend to escalate.