Viral Meningitis: This condition is characterized by inflammation of the meninges (the protective membranes covering the brain and spinal cord) due to a viral infection. Symptoms such as severe headache, fever, and vomiting are common, and while cough is not a primary symptom, it can occur due to the viral etiology. The high fever (104°F) and recurrent nature suggest a severe infection.

Bacterial Meningitis: Although less common than viral meningitis, bacterial meningitis is a serious condition that presents with similar symptoms, including severe headache, high fever, and vomiting. The presence of a cough could be indicative of a respiratory infection preceding the meningitis.
Influenza with Complications: Influenza (the flu) can lead to secondary bacterial infections such as pneumonia, which could explain the cough. Severe cases of the flu can also present with high fever, vomiting, and headache.
Pneumonia: Pneumonia, especially when caused by bacteria or viruses, can present with cough, high fever, and severe headache due to the infection and possible spread to the bloodstream (sepsis).

Subarachnoid Hemorrhage: Although this condition typically presents with a sudden, severe headache (often described as "the worst headache of my life"), it's crucial not to miss due to its high mortality rate. Fever and vomiting can occur, but cough would be an atypical symptom.
Encephalitis: An inflammation of the brain tissue, usually caused by viral infections. Symptoms can include fever, headache, confusion, and sometimes cough if the infection started in the respiratory system.
Sepsis: A life-threatening condition that arises when the body's response to infection causes injury to its own tissues and organs. Sepsis can present with fever, vomiting, and headache, among other symptoms. A cough could be present if the sepsis originated from a respiratory infection.

Brain Abscess: A rare but serious condition where a pocket of pus forms in the brain, usually as a result of a bacterial or fungal infection. Symptoms can include headache, fever, vomiting, and sometimes cough if the abscess is related to an otogenic (ear) or sinus infection.
West Nile Virus: A viral infection transmitted by mosquitoes, which can cause symptoms ranging from mild fever to severe neurological diseases like meningitis or encephalitis. While rare, it's an important consideration in endemic areas, especially during mosquito season.
Tuberculous Meningitis: A form of meningitis caused by Mycobacterium tuberculosis. It's rare in regions with good healthcare but can present with chronic symptoms including headache, fever, and vomiting. A cough could be present if there's concurrent pulmonary tuberculosis.