Some of this is a little hard to define, because the divisions of public/private in monarchies is not wholly analogous to the way we consider the public/private divide today.

A king's wealth is derived from taxation of his subjects, but it was hardly subject to the strict review, representative assessment, auditing, etc., that we expect from public funds in present context. To use a turn of phrase familiar in American contexts, there was "taxation without representation," at least not of an elected nature. Religious institutions also [taxed] (https://books.google.ca/books?id=N37WCgAAQBAJ&pg=PA65&lpg=PA65&dq=religious+institutions+taxing+in+middle+ages&source=bl&ots=da80exR97g&sig=J38FYAU0UHNEHkKuRru9mRd-Ybk&hl=en&sa=X&ved=0ahUKEwiLkLnLjdbPAhXJ64MKHZlLCPoQ6AEINzAF#v=onepage&q=religious%20institutions%20taxing%20in%20middle%20ages&f=false) their adherents. In Muslim Caliphates, "people of the book" (ie., Christians and Jews) had to pay additional taxes.

Likewise for other members of landed gentry-- they're hardly "independently wealthy" if their income is derived from taxes paid on land which the dwellers are obligated to pay, but their role is not one of a private donor either. (In the UK, you still have to pay taxes to various members of the landed gentry for the privilege of living on the land. In London, many people pay to the [Duke of Westminster] (https://en.wikipedia.org/wiki/Hugh_Grosvenor,_7th_Duke_of_Westminster) -- the Duke is the richest person on earth under 30. Anyway, that's a story for another day.)

The important thing to appreciate is that conceptions of the "public purse" in the strictest sense are fairly modern. Kings, queens, feudal lords, and religious institutions had tax capacity which is not consistent with modern conceptions of tax.

Still, regardless of how we refer to them, the funding of exploration was largely the purview of nobility, as one might anticipate due to their wealth. For instance, Christopher Columbus' voyages to "India" were financed by Isabella I of Castile. The story is often told in a somewhat exaggerated way as the passage I cite below mentions, but she was a major creditor.

Isabella I was the queen of Castile and Aragon, precursors to modern Spain, and was adamant that exploration of the New World serve her broader ambition of [Christianizing.] (https://www.britannica.com/biography/Isabella-I-queen-of-Spain) Her motivations for funding the exploration were of a deeply evangelical nature (in the classical sense of the word.)

In addition to her funding of Columbus' voyages, she also expelled Jews and Muslims living in the region, to further ambitions of having a wholly Christian nation:

Although the story of her offering to pledge her jewels to help finance the expedition cannot be accepted, and Columbus secured only limited financial support from her, Isabella and her councillors must receive credit for making the decision to approve the momentous voyage. The terms on which the expedition was to set out to discover a new route to the Indies were drawn up on April 17, 1492. The New World that was explored as a result of that decision was, with papal confirmation, annexed to the crown of Castile, in accordance with existing practice in regard to such previous Atlantic discoveries as the Canary Islands.

The queen and her advisers hardly needed Columbus to remind them of the opportunity now offered for the spreading of Christianity. Yet the unexpected discoveries quickly brought fresh problems to Isabella, not the least of which was the relationship between the newly discovered “Indians” and the crown of Castile. The queen and her councillors were more ready to recognize the rights of the Indians than was Columbus; she ordered some of those he had brought back as slaves to be released. The queen was still concerned with these problems when she died in 1504.

Meanwhile, in 1480 the Inquisition had been set up in Andalusia. There is little doubt that this represented the culmination of a long and popular movement against non-Christians and doubtful converts, which had manifested itself frequently in the late Middle Ages in Castile. The expulsion in 1492 of those Jews who refused conversion was the logical result of the establishment of the Inquisition. Yet, however meritorious the expulsion may have seemed at the time in order to achieve greater religious and political unity, judged by its economic consequences alone, the loss of this valuable element in Spanish society was a serious mistake.

There are other examples, of course, but Isabel I is a very famous case of a monarch funding exploration of the New World.

For more information you can also check out: Liss, Peggy K. Isabel the Queen: Life and Times. New York, N.Y.: Oxford University Press, 1992