The "Mad King" most famously refers to Aerys II Targaryen from the fictional world of "Game of Thrones" and "A Song of Ice and Fire." Aerys II was the seventeenth and last Targaryen king to rule Westeros, known for his descent into paranoia, cruelty, and madness, which led to violent acts and his eventual assassination by Jaime Lannister during Robert's Rebellion. His madness was influenced by various personal and political tragedies, including the loss of his children and imprisonment during a rebellion. This character is central to the background story of "Game of Thrones" and the precursor events shown in "House of the Dragon".
There are other historical figures known as the "Mad King," such as Ludwig II of Bavaria, who reigned in the 19th century and was known for his artistic patronage and eventual declaration of insanity, and Charles VI of France, who suffered from severe mental illness during his reign in the late 14th and early 15th centuries.
If the interest is in the fictional "Mad King," it is Aerys II Targaryen. For historical contexts, Ludwig II of Bavaria and Charles VI of France are notable as well.