Gemara

45a (Mishnah): We do not redeem captives for more than their value (to be sold for slaves). This is a decree for Tikun ha'Olam (to fix matters).

Question: Is this to avoid depletion of the community's money, or lest captors be aroused (to strive to capture more people, or to demand more money)?

Answer: Levi bar Darga redeemed his daughter for 13,000 gold coins. (We are not concerned for inciting captors, only for the Tzibur's money.)

Rejection (Abaye): Perhaps Chachamim disapproved of his action!

58a (Beraisa): In Rome, R. Yehoshua ben Chananyah heard about a beautiful child who was a captive. He stood at the entrance of the prison and said "who caused Yakov to be despoiled, that Yisrael should be to the plunderers?"

The child: "For we have sinned to Hash-m, and did not desire to go in His ways, and did not listen to His Torah."

R. Yehoshua: I am sure that this child will give rulings in Halachah! I will not leave until I redeem him for anything they ask. He redeemed him for a fortune. After a short time, the child, R. Yishmael ben Elisha, gave a ruling in Halachah.

Kesuvos 52a (Beraisa #1): If captors demand up to 10 times a woman's value, the first time, her husband must redeem her. After this, it is his choice;

R. Shimon ben Gamliel says, we do not redeem captives for more than their value. This is an enactment for Tikun ha'Olam.

Inference: Up to her value we redeem, even if it exceeds her Kesuvah!

Rishonim

The Rif (Kesuvos 19a) brings the Beraisos in Kesuvos, and rules like R. Shimon (ben Gamliel).

Rosh (Kesuvos 4:22): The Rif rules like R. Shimon because the Stam Mishnah in Gitin is like him. The Mishnah discusses captives Stam, even one's wife. The Ramah rules like the first Tana, for they are the majority. The Mishnah in Gitin discusses other captives. One's wife is like himself. Just like one may give all his money to redeem himself, also for one's wife. This seems correct.

Rambam (Hilchos Matanos Aniyim 8:12): We do not redeem captives for more than their value. This is for Tikun ha'Olam, lest the enemy exert excessively to capture people.

Rosh (4:44): The Gemara did not resolve the reason for the enactment. A Beraisa obligates redeeming one's wife for 10 times her value. One's wife is like himself, more than his daughter is. There was no enactment not to give all one's money to redeem himself. Similarly, there was no enactment to limit redemption of a Chacham. R. Yehoshua redeemed a child for a great sum, for he saw that he was exceedingly sharp and had potential to be great. All the more so we do so for one who is already great!

Tosfos (45a DH d'Lo): We cannot bring a proof from R. Yehoshua, for at the time of the Churban, in any case the enemy captured wantonly.

Note: Also, perhaps this was before the enactment!

Ran (Gitin 23a DH v'Ein): Some say that even though the question was not settled here, it is settled from the Beraisa in Kesuvos. R. Shimon forbids redeeming for more than her value, even when her husband redeems her and there is no concern for the Tzibur's money. We cannot say that the first Tana agrees to our Mishnah, but he holds that the enactment was lest we deplete the Tzibur's money. The text of our Mishnah is just like R. Shimon's words. Surely, our Tana forbids for the same reason. The first Tana there argues with our Mishnah. Often, a question is unsettled, and is settled elsewhere. R. Yehoshua redeemed the child. for he recognized that he is a great Chacham. No one is like a Chacham. We are not concerned lest this incite captors. We redeem him for any price in the world. He has priority over Melech Yisrael (Horayos 13a). I say that the question was not settled there. If it were, our Gemara would have brought the Beraisa! There is concern for depleting the Tzibur's money, and all the more so for depleting the husband's money. We should not obligate him to redeem her for more than her value! The Rishonim held that since the question was not settled, and there is concern for Takalah, we are passive not to redeem. Also, we proved from Levi bar Darga that one may redeem for any amount. We do not abandon this due to Abaye's doubt.

Hagahas Yam Shel Shlomo: The last words of the Ran refer back to his initial opinion, that the question was not settled, and perhaps the concern is depleting the Tzibur's money.

Bach (YD 252:3): The Ran holds that the Tzibur does not redeem people for more than their value, and one need not do so for his wife, but one may redeem her or anyone for as much as he wants. The Beis Yosef understood that the Ran forbids redeeming anyone for more than his value, like the Rif. This is wrong.

Poskim

Shulchan Aruch (YD 252:4): We do not redeem captives for more than their value. This is for Tikun ha'Olam, lest the enemy exert excessively to capture people. One may redeem himself for as much as he wants. Similarly, one may spend much to redeem a Chacham, or even a sharp Talmid who is not a Chacham but has potential to be great.

Pischei Teshuvah (4): Eshel Avraham says that Nachalas Shiv'ah proves that when there is concern lest captives assimilate or be killed, we redeem without concern for the Tzibur's money. Yad Eliyahu says that even then we do not redeem for more than the value. Beis Hillel says that we do when there is mortal danger, for one who neglects to redeem transgresses "Lo Sa'amod Al Dam Re'echa." One must forfeit all his money to avoid transgressing a Lav (OC Sof 656). This is no proof, for this refers only to a Lav with an action (Pri Megadim 656:8 and Chavos Ya'ir 139). Keneses Yechezkeil (38) says that since the Rambam and Shulchan Aruch are concerned for inciting captors, we do not redeem for more than the value even when there is mortal danger. A case occurred in which a Yisrael was accused of Zenus with a Nochris; Maharam Lublin (15) said that they should pay more than his value, to avoid Chilul Hash-m. Perhaps we can say "we hold like the Rif and Rosh, that the question was not settled.' In places where they do not capture for the sake of money, perhaps the Tzibur is obligated to redeem for any price when there is mortal danger.

Pischei Teshuvah (5): Maharam Lublin (15) says that even though there is no slave market here, we assess the price in places where there is. The Radvaz (Chadashos 1:40) says that we follow the redemption price of Nochrim, but justifies the practice to pay more and says that it is a great Mitzvah.

Rema: See the Tur EH 78. (He says that some, including the Rosh, obligate to redeem one's wife even for more than her value, but the Rif and Rambam do not obligate for more than her value.)