In the first row of the remarkably preserved Jewish cemetery in Worms (one we will please G-d visit on this summer's Journey through Jewish History) one will see gravesite of the Rav Meir the son of Baruch, the Maharam of Ruteneberg, the leading Rabbinic figure in Germany in the 13th Germany and considered to be the last of the Tosasfists. Immediately next to him is the grave of Alexander ben Salomon Wimpfen the man who redeemed Rav Meir's body from captivity.
In 1286 "a new king arose in Germany who knew not Rav Meir." The Maharam was captured as he tried to flee Germany[1]. While the Jewish community raised the exorbitant ransom demanded by the German authorities, Rav Meir refused to be redeemed in keeping with the ruling of the Mishna in Gittin that "one may not redeem captives for more than their monetary value." (Gittin 45a) He thus spent the last seven years of his life in captivity and it was not until seven years after his death when Alexander Wimpfen paid a ransom that his body was released. Upon his own death that very same year he was buried immediately next to the Maharam. Visiting  their graves over 700 years later is most moving and powerful. 
The question of ransoming captives is one that is sadly all too relevant. Initially one could, and presumably many did, pay whatever was demanded. As the Mishna makes clear it was only a rabbinic edict, mifnei tikkun olam, to better the world as a whole, that the Sages forbade the paying of excessive ransoms. While the results might be unfortunate for the captive at hand for the world at large this was a necessary edict. 

The Gemara wonders what the main motivating factor was for this decree. Is it that giving in to captors encourages more hostage taking and while a fair ransom could be paid an excessive one could not or is it primarily for economic reasons that we are not allowed to pay excessive ransoms? Doing so is liable to bankrupt the community, something we cannot do for the sake of hostages. That may sound harsh but given the choice we must sacrifice the few for the benefit of the many.
There is a crucial difference between these two reasons; if the reason is communal affordability then family members or friends may use their own money to redeem the hostage[2], something we would not allow if the fear is giving in to terror. Interestingly the Talmud does not reach a conclusion. The extremely brief discussion on this topic concludes with the case of Levi the son of Darga who paid 13,000 golden dinars for the release of his daughter, implying the issue is the financial cost to the community. However this case is dismissed as evidence as perhaps Levi ben Darga "acted against the will of the Sages." What a parent will do for a child cannot be brought as evidence for people in general. 
Furthermore the Talmud makes no distinction between different types of hostage situations leaving us wondering if this same limitation applies in a case where the life of the hostage is in danger. The Tosafists (Gittin 58a s.v. kol) take it as a given that our Mishna refers only to cases of monetary extortion - as was the case by the Maharam of Rutenberg. If the hostage's life is in danger then obviously we pay even exorbitant sums to save someone's life. Overpaying for hostages is not one of the three cardinal sins that require martyrdom. 
Yet not all agree. If the goal is to prevent future hostage taking or the financial stability of the community it matters little under what threat the hostages find themselves. Presumably one would pay even more to ransom a hostage facing possible death making future hostage taking even more likely and undermining any possible tikkun olam. Thus many of the commentaries make no distinction and rule in all cases that one may not redeem a captive for more than their value. 
But what about pikuach nefesh? Are we going to let someone die because of lack of funds? The simple answer is yes and we do it all the time. Pikuach nefesh is fundamentally an individual obligation. One must do all one can to save one's life. And whether we are afraid of encouraging hostages or financial stability all authorities agree that an individual may pay any amount to redeem himself and their spouse (though not a child[3]). However a community must take a much broader approach, one that benefits the most amount of people. It is for this reason governments can and must spend money on education, transportation and even museums and opera houses, monies that could be spent on hospitals and health care. The decree not to overpay for hostages was initiated mifnei tikkum olam for the benefit of the world at large even as for certain individuals the result is much worse. 
It is through this prism that we can understand the concept of "paying more than their value." To put a monetary value on a life sounds cruel and in many ways is. Every human life is priceless and "he who saves one life is as if he has saved an entire world." (Sanhedrin 37a) Yet in making public policy decisions we must make a cost benefit analysis even if the cost might be measured in human lives.
There is a third approach, one that distinguishes between the reasons for the limitation. If the concern is lest it encourage future hostage taking then the decree not to "overpay" must apply even if the hostage's life is in danger. If however the fear is the cost to the community then if a person's life is in danger we will spend whatever needs to be spent to save a life. 
What best reflects tikkun olam, the overarching benefit is not always easy to determine. For some it is it saving one life now even at the possible risk to many later whereas others feel we are better off sacrificing the few now to better protect the many later. Sadly this debate is not only theoretical and applying it to real-life situations is both difficult and necessary. 
[1] His pre-eminent student Rabbeinu Asher (The Rosh) did eventually leave Germany and in 1303 was appointed Rabbi in Toledo, Spain (another place we will visit on this summer's Journey through Jewish History) leading to the unique situation of a great Ashkenazic Rabbi leading a Sephardic community. 
[2] To the best of my knowledge no Western government would allow family members to pay a ransom to kidnappers.
[3] While we love our children more than ourselves they - as opposed to spouses, who are considered as one - are independent of us. As we read this past Shabbat in parshat Mishpatim, while we are responsible for damages caused by our animals we are not liable for those caused by our children.