I assume that your story is supposed to be set in Eruope roughly about 1500 instead of in Asia, Africa, or the Americas.
I note that if you describe your boy precisely, that may be offputting to some readers. If you describe your boy rather vaguely, different readers will imagine him differently, and if your story become popular, might draw him differently.
One fact about humans is that they grow at varying rates. At one extreme someone can be a child in age and a man in size and physical development, & at another extreme someone can be a man in age and a child in size and physical development.
So possibly some readers will imagine that your boy is much bigger and stronger than other readers imagine him. And I don't know if it would be a good idea to clearly describe how big and strong he is.
If you boy has a powerful enough bow, and is strong enough to use it, and experienced in using it, he could hunt just about anything, or perhaps accidentially encounter just about anything when hunting something smaller and safer, and be forced to shoot the unexpected critter to save his own life.
People do hunt elephants successfully with bows, and on the other hand professional elephant hunting, even using elephant guns, was or is a very dangerous job with a high fatality rate.
So it is theoretically possible for your boy character to hunt and kill just about any land animal. The larger and more dangerous the animal, the greater the probability that he would eventually be killed by one of his prey animals.
Part One: Wolves
The palace and town of Versailles, 12 miles from Paris, was based on a hunting lodge built by King Louis XIII, father of Louis XIV, in 1623, and replaced with a small chateau in 1631-1634.
I think that the favorite prey around Versailles was wolves. Louis the Dauphin, son of Louis XIV, loved wolf hunting and is said to have killed a thousand wolves. other members of the royal family, including females, also hunted wolves.
If your story is set about AD 1500, there should be wolves almost everywhere in Europe.
Europe, excluding Russia, Belarus and Ukraine, has 17,000 wolves in more than 28 countries. In many countries of the European Union, the wolf is strictly protected under the 1979 Berne Convention on the Conservation of European Wildlife and Natural Habitats (Appendix II) and the 1992 Council Directive 92/43/EEC on the Conservation of Natural Habitats and of Wild Fauna and Flora (Annex II and IV). There is extensive legal protection in many European countries, although there are national exceptions.
Wolves have been persecuted in Europe for centuries, having been exterminated in Great Britain by 1684, in Ireland by 1770, in Central Europe by 1899, in France by the 1930s, and in much of Scandinavia by the early 1970s. They continued to survive in parts of Finland, Eastern Europe and Southern Europe. Since 1980, European wolves have rebounded and expanded into parts of their former range. The decline of the traditional pastoral and rural economies seems to have ended the need to exterminate the wolf in parts of Europe. As of 2016, estimates of wolf numbers include: 4,000 in the Balkans, 3,460–3,849 in the Carpathian Mountains, 1,700–2,240 in the Baltic states, 1,100–2,400 in the Italian peninsula, and around 2,500 in the northwest Iberian peninsula as of 2007.
From wolf wikipedia page.
Part Two: Aurochs
The aurochs (Bos primigenius) (/ˈɔːrɒks/ or /ˈaʊrɒks/) is an extinct cattle species, considered to be the wild ancestor of modern domestic cattle. With a shoulder height of up to 180 cm (71 in) in bulls and 155 cm (61 in) in cows, it was one of the largest herbivores in the Holocene; it had massive elongated and broad horns that reached 80 cm (31 in) in length.
From aurochs wikipedia page.
It was still widespread in Europe during the time of the Roman Empire, when it was widely popular as a battle beast in Roman amphitheatres. Excessive hunting began and continued until it was nearly extinct. By the 13th century, the aurochs existed only in small numbers in Eastern Europe, and hunting it became a privilege of nobles and later royals. Fossils found in West Bengal indicate that the Indian aurochs may have survived until the early 12th century.
The gradual extinction of the aurochs in Central Europe was concurrent with the clearcutting of large forest tracts between the 9th and 12th centuries. The population in Hungary declined since at least the 9th century and was extinct in the 13th century. Subfossil data indicate that it survived in northwestern Transylvania (in Romania) until the 14th to 16th century, in western Moldavia (also in Romania) until probably the early 17th century, and in northeastern Bulgaria and around Sofia until the 17th century at most. An aurochs horn found at a medieval site in Sofia indicates that it survived in western Bulgaria until the second half of the 17th to the first half of the 18th century.
The last known aurochs herd lived in a marshy woodland in Poland's Jaktorów Forest. It decreased from around 50 individuals in the mid 16th century to four individuals by 1601. The last aurochs cow died in 1627 from natural causes.
Aurochs extinction section.
So your boy could hunt an auroch if the story is set in eastern Europe.
Part Three: Bison
The European bison (Bison bonasus) or the European wood bison, also known as the wisent[a] (/ˈviːzənt/ or /ˈwiːzənt/), the zubr[b] (/zuːbər/), or sometimes colloquially as the European buffalo,[c] is a European species of bison. It is one of two extant species of bison, alongside the American bison. The European bison is the heaviest wild land animal in Europe, and individuals in the past may have been even larger than their modern-day descendants. During late antiquity and the Middle Ages, bison became extinct in much of Europe and Asia, surviving into the 20th century only in northern-central Europe and the northern Caucasus Mountains. During the early years of the 20th century, bison were hunted to extinction in the wild.
In 1513 the Białowieża Forest, at this point one of the last areas on Earth where the European bison still roamed free, was transferred from the Troki Voivodeship of Lithuania to the Podlaskie Voivodeship, which after the Union of Lublin became part of the Polish Crown. In the Polish–Lithuanian Commonwealth, at first European bison in the Białowieża Forest were legally the property of the Grand Dukes of Lithuania and later belonged to the Crown of the Kingdom of Poland. Polish-Lithuanian rulers took measures to protect the European bison, such as King Sigismund II Augustus who instituted the death penalty for poaching bison in Białowieża in the mid-16th century. Wild European bison herds existed in the forest until the mid-17th century. In 1701, King Augustus II the Strong greatly increased protection over the forest; the first written sources mentioning the use of some forest meadows for the production of winter fodder for the bison come from this period. In the early 19th century, after the partitions of the Polish Commonwealth, the Russian tsars retained old Polish-Lithuanian laws protecting the European bison herd in Białowieża but also turned all the people inhabiting the area into serfs. Despite these measures and others, the European bison population continued to decline over the following century, with only Białowieża and Northern Caucasus populations surviving into the 20th century. The last European bison in Transylvania died in 1790.
From European bison's wikipedia page
So if your story is set in eastern Europe, your boy could hunt European bison, and possibly face the death penalty if caught.
Part Four: Moose.
As a child, I was shocked reading Bambi by a scene where Bambi gazed in awe at passing Elk, because the artist drew the Elk looking like moose instead of American elk.
The moose (in North America) or elk (in Eurasia) (Alces alces) is a member of the New World deer subfamily and is the largest and heaviest extant species in the deer family.
In Europe, moose are currently found in large numbers throughout Norway, Sweden, Finland, Latvia, Estonia, Poland, with more modest numbers in the southern Czech Republic, Belarus, and northern Ukraine. They are also widespread through Russia on up through the borders with Finland south towards the border with Estonia, Belarus and Ukraine and stretching far away eastwards to the Yenisei River in Siberia. The European moose was native to most temperate areas with suitable habitat on the continent and even Scotland from the end of the last Ice Age, as Europe had a mix of temperate boreal and deciduous forest. Up through Classical times, the species was certainly thriving in both Gaul and Magna Germania, as it appears in military and hunting accounts of the age. However, as the Roman era faded into medieval times, the beast slowly disappeared: soon after the reign of Charlemagne, the moose disappeared from France, where its range extended from Normandy in the north to the Pyrenees in the south. Farther east, it survived in Alsace and the Netherlands until the 9th century as the marshlands in the latter were drained and the forests were cleared away for feudal lands in the former. It was gone from Switzerland by the year 1000, from the western Czech Republic by 1300, from Mecklenburg in Germany by c. 1600, and from Hungary and the Caucasus since the 18th and 19th century, respectively.
From moose wikipedia page.
So if your boy hunts moose, the story should be set in northern or eastern Europe.
Part Five: Brown Bear
The brown bear (Ursus arctos) is a large bear species found across Eurasia and North America.
In Europe, in 2010, there were 14,000 brown bears in ten fragmented populations, from Spain (estimated at only 20–25 animals in the Pyrenees in 2010, in a range shared between Spain, France and Andorra, and some 210 animals in Asturias, Cantabria, Galicia and León, in the Picos de Europa and adjacent areas in 2013) in the west, to Russia in the east, and from Sweden and Finland in the north to Romania (5000–6000), Bulgaria (900–1200), Slovakia (with about 600–800 animals), Slovenia (500–700 animals) and Greece (with about 200 animals) in the south.
From brown bear wikipedia page
Bears became extinct in Britain about 1,000 years before the time of your story, but they could be found all over continental Europe around 1500, though probably always far from cities and towns.
Emperor Louis IV (1282-1347) died of a heart attack near Munich, Germany on 11 October 1347 aged 65 years, 6 months, and 10 days. He was on a bear hunt. I guess if it was considered safe enough for an elderly emperor to hunt bears, it would be alright for a 15-year-old boy, if he was big and strong, to hunt bears with spears. And if the boy is smaller and weaker, he might hunt bears with bow and arrows.
Part Six: Wild Boar
The wild boar (Sus scrofa), also known as the wild swine, common wild pig, Eurasian wild pig, or simply wild pig, is a suid native to much of Eurasia and North Africa, and has been introduced to the Americas and Oceania.
There are many myths and legends about heroes hunting wild boars, which often seem about as large and dangerous as buffalo or elephants in such tall tales. And wild boars can be quite ferocious and dangerous in real life. Boar spears are made with cross guards near the blade to stop speared boars from pushing along the length of the spear and getting to the person who speared them.
Wild boars once ranged in most of Eurasia.
In recent centuries, the range of wild boar has changed dramatically, largely due to hunting by humans and more recently because of captive wild boar escaping into the wild. Prior to the 20th century, boar populations had declined in numerous areas, with British populations probably becoming extinct during the 13th century. In the warm period after the ice age, wild boar lived in the southern parts of Sweden and Norway and north of Lake Ladoga in Karelia. It was previously thought that the species did not live in Finland during prehistory because no prehistoric wild boar bones had been found within the borders of the country. It was not until 2013, when a wild boar bone was found in Askola, that the species was found to have lived in Finland more than 8,000 years ago. It is believed, however, that man prevented its establishment by hunting. In Denmark, the last boar was shot at the beginning of the 19th century, and by 1900 they were absent in Tunisia and Sudan and large areas of Germany, Austria and Italy. In Russia, they were extirpated in wide areas by the 1930s. The last boar in Egypt reportedly died on 20 December 1912 in the Giza Zoo, with wild populations having disappeared by 1894–1902. Prince Kamal el Dine Hussein attempted to repopulate Wadi El Natrun with boars of Hungarian stock, but they were quickly exterminated by poachers.
From Wild boar wikipedia page
So your boy should be able to encounter wild boars in most parts of continental Europe about AD 1500.
Part Seven: Red Deer
The red deer (Cervus elaphus) is one of the largest deer species. A male red deer is called a stag or hart, and a female is called a hind. The red deer inhabits most of Europe, the Caucasus Mountains region, Anatolia, Iran, and parts of western Asia.
Red deer were probably found all over Europe about 1500.
In most of Euorpe nobles formed deer preserves or parks for hunting red deer. They hunted red deer for sport, for practice for war, and for food, thus often eating more meat than peasants. And there were often laws against commoners hunting deer, sometimes punished with the death penalty.
My lord Enguerrand de Couci had, as a nobleman, had hanged three noble youths (who were, as has been said, with the Abbot of St. Nicholas-aux-Bois in the diocese of Laon), because they were found in his woods with bows and arrows but without dogs or any equipment (engins) for catching wild beasts. The said abbot and some women cousins of those hanged carried a complaint concerning their deaths before the king.
From Trial of Enguerrand IV, Lord of Coucy, 1259.
And in this case it is not specified that the boys were hunting deer - I have seen another account in which they were hunting hares.
So depending on the social rank of your boy character, deer could be the most dangerous prey he could hunt in Europe about AD 1500.