We are, of course, talking institutionally, rather than personally - because everyone going into a battle, no matter how assured one is of victory, is bound to feel fear.

That said, there were a few places and situations that - while it's hard to say that the Mongols were "afraid", they did eventually give up on or steer clear of. These situations all share a common element: the Mongols didn't do well when their primary tactic of cavalry-charge was negated.

First and probably most famously, Kublai Khan's two attempts to conquer Japan. The Mongols had by this point conquered Korea, 2/3 Chinas, and almost the totality of Asia - how hard could a few backward, underpopulated pirate islands be?

Turns out: pretty hard.

In 1266 Kublai sent a message to the court of Emperor Hameyama saying:

Cherished by the Mandate of Heaven, the Great Mongol emperor sends this letter to the king of Japan. [...] Our relation is feudatory like a father and son. We think you already know this. Goryeo is my eastern tributary. Japan was allied with Goryeo and sometimes with China since the founding of your country; however, Japan has never dispatched ambassadors since my ascending the throne. We are afraid that the Kingdom is yet to know this. Hence we dispatched a mission with our letter particularly expressing our wishes. Enter into friendly relations with each other from now on. We think all countries belong to one family. How are we in the right, unless we comprehend this? Nobody would wish to resort to arms.

The Shogun, Hojo Tokimune, declined to issue a response - which was in and of itself essentially saying "come at me, bro."

The first invasion would set out in 1274. The delay was brought about by the fact that the Khan had no navy capable of ferrying his expedition force, and so one had to be built first. The force consisted of some 15,000 Mongol and Chinese troops, along with 8,000 Korean troops in 300 ships, along with 4-500 smaller craft. The force took the islands of Tsushima and Iki and then landed at Hakata on the eastern coast of Kyushu.

Though the Japanese forces had prepared considerable shoreline defenses at Hakata, the Mongol/Chinese weaponry and tactics (including long distance weapons like crossbows and catapults to which the Japanese had no counter) turned the early battle in their favor. However, it was late November, and the weather would step in to quite literally turn the tide. From Rossabi:

unexpectedly a gale-force storm struck that night. The Japanese were accustomed to such "unnatural" occurrences and simply took cover in whatever shelter they could find. But the Mongols were terrified and were persuaded by their Korean underlings to return to their ships and head for the open seas until the storm subsided. The results were disastrous: The winds, waves, and rocks shattered a few hundred ships, and about thirteen thousand lives were lost. The expedition thus ended in disaster for the Mongols, and the remaining forces sailed for home and reported on the fiasco to Khubilai.

At the time, Kublai was too tied down in the final push against the Southern Song Dynasty to be able to immediately launch another invasion force against the little island that could. Instead, he sent yet another emissary to the imperial court once again demanding Japanese submission. This time, however, rather than sending the emissary away empty handed as before, the Japanese court had the messenger executed.

It would be some 7 years before Kublai had finally conquered the Song, nd could turn his full attention back to the Japanese islands that yet defied him. In 1281, he launched his second invasion of the archipelago, this time not with some paltry for of 15,000... but with firepower of his fully armed and operational battlestation a reported 100,000 trooped, 15,000 sailors, and 900 ships. This time the Khan meant business.

The northern half of the expedition, some 40,000 troops, used a route very similar to the 1274 invasion force, first occupying Iki and then using that as a staging platform toward Kyushu. The southern half had been held up in the south, and so would link up with the northerners later one, once they'd fully mustered at Ningbo. Yet within the ranks, morale was... rather tense between the Mongol commanders, the Chinese troops, and the Korean sailors. Who'd've thunk? Again from Rossabi:

The Chinese troops, who comprised the vast majority of the soldiers, had scarcely any stake in the combat and so did not exert themselves to put up a strong fight. Moreover, when they landed in Kyushu, they were afforded no protection from either their enemies or the elements. They had to camp out in the open without a castle, stronghold, or town that would provide adequate defense and from which they could make forays against the enemy. The two armies fought for almost two months, but no clear victor emerged.

By this point, August had rolled around which - as everyone living on the Eastern Pacific coast well knows - is prime typhoon season. On Aug. 15th, a typhoon rolled over Kyushu at full-steam reportedly killing almost half of the assembled mongol force then and there. For the Japanese it was a miracle - the gods had sent forth this wind to protect them from the barbaric hordes at their doorstep and saved Japan. And that is actually where the term kamikaze derives - the "divine wind" which would save the home islands from foreign conquest in the hour of greatest need.

For the Mongols, it had rather the opposite effect. Already noting that their troops were on somewhat shaky morale grounds, "For the Mongols, the defeat was a devastating shock. Their failure shattered their mantle of invincibility, as Khubilai's subjects now realized that the Mongols were vulnerable. One of the principal underpinnings of their power — the psychological edge of terror that they held over their opponents — was badly shaken, if not destroyed."

But the hits just came on coming for Kublai and the Mongols. Southeast Asia likewise proved a complete debacle for the Mongols. A 1277 expedition to the kingdom of Pagan (mod. Burma) failed to capture the rebellious King Narathihapate, who had incensed the Khan when he'd killed three Mongol emissaries who had arrived demanding (what else?) unconditional submission. And seaborne expedition to from Guangdong to Champa (S. Vietnam) to subdue its King Jaya Indravarman VI met with total failure in the face of the guerrilla-style jungle tactics the Champa employed. Kublai then sought to commit a much larger overland force, but that would have to cross the Kingdom of Annam (N. Vietnam). While the Annam king, Tran Than-ton, had been paying lip-service to the Mongol court, he was not willing to allow a full-scale invasion force march through his territory. Though he'd been a nominal ally, when Kublai tried to force the issue by sending his General Sodu and his son Prince Toghon at the head of a Mongol column in 1285, a combination of guerrilla fighting, heat, and tropical disease quickly laid waste to the Mongol army, and resulted in the death of General Sodu (and something like 75-90% of the total force). Though Kublai would ultimately succeed in overthrowing the Pagan king, Narathihapate, it wouldn't be until 1287 and two further failed attempts. And for what? So that the newly installed Pagan government would pay lip-service to the throne.

Finally, the disaster at Java. In 1289, the Javanese king, Kertanagara, enraged the Khan when he branded the face of a Mongol emissary. Having apparently not learned the lesson of Japan that "Mongols + Ocean = Disaster", he once again organized a large fleet to punish this impudent island king. In 1292, the fleet set out and arrived at Java early the following year. Though the early encounters went well enough - so well, in fact, that the offending king was deposed by his own people, and his son-in-law Prince Vijaya assumed command of the Javanese. Vijaya offered terms of submission, but then betrayed the Mongols, ambushed their main camp at night, and killed a sizable majority of the force. The remainder of the expedition, now with no hope of success due to their diminished numbers, fled to the coast to await transport back to the mainland. However, due to seasonal shifts in the monsoon climate, they were essentially marrooned for a further 6 months, as no Mongol ship could reach the islands until the winds picked back up. The force would be picked off little by little over the following half year until, utter demoralized and sapped of strength, they were finally able to abandon the failed and enormously costly expedition altogether.

#So, to sum up:

Did the Mongol actually "fear" anyone? Unknown.

But we can say there are two things that they very well should have feared and avoided: Water and jungles.

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Rossabi, Morris. "The reign of Khubilai khan" in The Cambridge History of China, vol. 6: Alien Regimes and Border States.

Turnbull, Stephen R. Genghis Khan and the Mongol Conquests, 1190–1400.