When Duryodhana was taking the last breath in the field of Kurukshetra, at that time he had raised his three fingers. Lord Krishna went to him and understood that Duryodhana wants to say that if he did not do three mistakes in the war then he would win the war. But Krishna told Duryodhan that if you do anything then you will lose. After hearing this, Duryodhan lowered his finger.
Which 3 mistakes did Duryodhan want to say?
No such event of raising 3 fingers happened between Duryodhana & Krishna.
After Duryodhana was stuck down with broken thighs, he angrily complained to Krishna about how his tactics of deceit had taken the lives of great car-warriors like Bhishma, Drona, Karna, Jayadratha, himself & so on.
Krishna replied by counting all the misadventures attempted by Duryodhana & co, like poisoning Bhima, burning them alive in house of lac, insulting Draupadi, deceiving Yudhishtira of his kingdom & so on.
However, Duryodhana being adamant to his own POV curses Krishna and that's followed by celestial beings praising Duryodhana with roars and flowers.
Duryodhana said, "... Who is there so fortunate as myself! ... With all my well-wishers, and my younger brothers, I am going to heaven, ... As regards yourselves, with your purposes unachieved and torn by grief, live ye in this unhappy world!"
Sanjaya continued, 'Upon the conclusion of these words of the intelligent king of the Kurus, a thick shower of fragrant flowers fell from the sky. The Gandharvas played upon many charming musical instruments. The Apsaras in a chorus sang the glory of king Duryodhana. The Siddhas uttered loud sound to the effect, "Praise be to king Duryodhana!" Fragrant and delicious breezes mildly blew on every side. All the quarters became clear and the firmament looked blue as the lapis lazuli. Beholding these exceedingly wonderful things and this worship offered to Duryodhana, the Pandavas headed by Vasudeva became ashamed. [Shalya Parva, section 61]
It looks like an anti-climax, but Krishna handled the situation by telling PAndava-s that, whatever they did under his command was the only way to win the war & that's was righteous in that context. Later they departed from the place.
From desire of doing good to you, I repeatedly applied my powers of illusion and caused them to be slain by diverse means in battle. If I had not adopted such deceitful ways in battle, victory would never have been yours, nor kingdom, nor wealth! ... You should not take it to heart that this foe of yours hath been slain deceitfully. When the number of one's foes becomes great, then destruction should be effected by contrivances and means. The gods themselves, in slaying the Asuras, have trod the same way. That way, therefore, that hath been trod by the gods, may be trod by all. We have been crowned with success. It is evening. We had better depart to our tents. ... All those kings, possessed of arms that resembled spiked bludgeons, then proceeded towards their tents, filled with joy and blowing their conchs on their way. [Shalya Parva, section 62]
Duryodhana did not raise three fingers. Duryodhana didn't die thinking about his regrets, he died happy about Ashwatthama killing the Upapandavas. Here is what this chapter of the Sauptika Parva of the Mahabharata says:
Duryodhana then, having heard those words that were so agreeable to his heart, regained his senses and said these words in reply, "That which neither Ganga's son, nor Karna, nor thy sire, could achieve, hath at last been achieved by thee today, accompanied by Kripa and Bhoja. Thou hast slain that low wretch (Dhrishtadyumna) who was commander of the Pandava forces, as also Shikhandi. In consequence of this I regard myself equal to Maghavat himself! Good be to you all! Let prosperity be yours! All of us will again meet together in heaven!" Having said these words the high-souled king of the Kurus became silent. Casting off his griefs for all his (slain) kinsmen, he then gave up his life-breath. His soul ascended to sacred heaven, while his body only remained on earth. Even thus, O king, thy son Duryodhana breathed his last. Having provoked the battle first, he was slain by his foes at last. The three heroes repeatedly embraced the king and gazed steadfastly on him.
So the story you're referring to is likely just folklore. It sounds similar to the story of Ravana having three unfulfilled desires in life: making ocean water drinkable, bulidjng a staircase to Swarga, and I forget the third. That story too is just folklore with no basis in Hindu scripture. It also sounds similar to the Sri Vaishnava Acharya Yamunacharya, who departed the Earth holding three fingers signifying his three unfulfilled desires in life: writing a Sri Vaishnava commentary on the Brahma Sutras, writing a Sri Vaishnava commentary on the Alwars' poems, and naming someone Parashara. Ramanujacharya fulfilled all three.
