Causes of Chronic, Constant Severe HeadacheSomatisation disorder: psychological problems may result in true or imaginary severe headache and other complaints
Temporal arteritis is an inflammation of the temporal artery. Symptoms include:
Chronic one-sided headache and extreme tenderness
Low-grade fever
Carotidynia is an inflammation of the carotid artery with (1):
One-sided throbbing pain and tenderness in the neck and head
Brain tumor or cyst symptoms may develop slowly over several months and may include:
Recurrent headaches, worse in the morning
Abnormal sensations and muscle weakness on one side of the body
Personality changes may develop slowly over several months
Slurred speech, vision disturbance or other neurological symptoms
Brain aneurysm, even if not ruptured, can cause:
Constant, moderate or severe headache
Nausea or vomiting
Blurred vision
Drooping eye

Severe Headache in Small ChildrenCauses of severe headache in small children include:

Citrullinemia - a genetic disorder of metabolism of amino acids. Symptoms include (2): 
Severe headache, sleepiness, poor appetite, seizures
Hydrocephalus – a collection of the fluid that surrounds the brain due to a birth defect or as a complication of an infection, tumor, trauma. Symptoms:
Rapidly growing head
Eyes that seem to gaze downward
Prominent fontanelles
Irritability
Constant severe headache