Rune Soldier Louie/WMG

Louie is the illegitimate son of the king of Ohfun and the high priestess of Mylee.
The king, the one time we see him, is almost a dead ringer for Louie, with the same build and hair. Louie calls the priestess "aunt", but she displays a clearly maternal interest in his life and well-being, and takes an especial delight in pairing him up with Melissa. The king, the priestess, and the head of the mage guild (with whom Louie was fostered) were all adventuring partners long before the start of the series. The king became the king not by birthright, but by a combination of conquest and a political marriage made around the time Louie was born. It seems obvious that Louie was fostered to facilitate the king's marriage, and to prevent both possible succession issues and any embarrassment to either the king or the high priestess.

The king probably doesn't know Louie is his son, judging by his reaction in the ending. If he shares Louie's recklessness, the high priestess might've purposely hidden it from him; if he knew, he might "do the right thing," acknowledge Louie as his son, and send the kingdom into turmoil.

Louie is the illegitimate son of Headmaster Carwes and High Priestess Jenny.
As noted above, High Priestess Jenny expresses an almost unmistakably maternal concern for Louie and seems very interested in his romantic life. She also has a cutesy pet name for him.

During the Mylee Festival, Louie notices a painting of her alongside Headmaster Carwes and remarks that she never told him that the two of them fought together in the war. She responds by smiling secretly and saying that she and headmaster Carwes are "quite close." It seems strange that both she and Headmaster Carwes would have neglected to tell Louie such an interesting piece of information, especially since Louie is both interested in and knowledgeable about the battle. They seem to be actively hiding the extent of their relationship; there must be a reason.

Headmaster Carwes's concern for Louie is understandable, as he is Louie's foster father; but watch his face at the end of the last episode when he begs the king not to "punish my son." We also know that Carwes was a beefy guy in his younger days - see the aforementioned painting.

One possible reason for Louie's fostering: we don't know if the priestesses of the Mylee shrine are allowed to marry or have families. It's an all-female religion, and there are no married priestesses or priestesses with children mentioned in the series. Odds are high that Priestess Jenny was unable to acknowledge her child because of the policies of her church and "fostered" Louie with his real father to avoid awkward questions.

Alternately, Louie is the illegitimate son of King Kashue of Flaim.
Kashue adventured extensively in Alecrast in his youth and may well have hooked up with Louie's mother (who also converted him to the worship of Myri, which is why it's apparently Flaim's state religion).

Alternately, Louie's parents are completely unknown people that never make an appearance.
In anime, it would be strange for an orphan not to be related to some other significant character; but it may be possible. For Louie to have lasted any amount of time in the Mage Guild, he would need to have some sort of connection. It would be only natural for the head of the Mage Guild to be at least casually associated with the High Priestess; given how she acts, it wouldn't be all that surprising for her to act that way towards him without an ulterior motive.

Louie is the son of Kenshiro.
Why do you think punching is his first answer to every conflict?