Viral Exanthem (e.g., Roseola, Measles, or Enteroviral infection): This is the most likely diagnosis due to the combination of fever, cough, and rash, which are common presentations of viral exanthems in children. The centripetal nature of the rash (starting centrally and spreading outward) is also consistent with some viral infections.

Pneumonia (bacterial or viral): The presence of fever and cough suggests a respiratory infection, and pneumonia could be a cause, especially if the rash is secondary to the infection.
Influenza: Influenza can present with fever, cough, and sometimes a rash, although the rash is less common.
Scarlet Fever: Caused by Group A beta-hemolytic streptococcus, it presents with fever, cough (less common), and a characteristic rash that could be described as centripetal, along with other symptoms like a "strawberry tongue."

Meningococcemia: Although less likely, this condition is life-threatening and can present with fever, rash (which can be petechial or purpuric and may appear centripetal), and cough (if there's an associated pneumonia). Early recognition is crucial for survival.
Rocky Mountain Spotted Fever (RMSF): This tick-borne illness can present with fever, headache, and a rash that typically starts centripetally. It's less common but critical to diagnose early due to its severity.
Kawasaki Disease: An acute febrile illness of childhood characterized by fever, rash, mucous membrane changes, and lymphadenopathy. It's a do-not-miss diagnosis due to the risk of coronary artery aneurysms if not treated promptly.

Erythema Multiforme: A skin condition characterized by target lesions, often triggered by infections or medications. It's less likely but could be considered if the rash has a distinctive targetoid appearance.
Juvenile Idiopathic Arthritis (JIA): Systemic JIA can present with fever, rash, and systemic symptoms, including cough if there's associated lung involvement. It's a diagnosis of exclusion and less common in the context provided.