Viagra medication is not physically addictive. However, it is possible to become psychologically dependent on these medications if not used in the appropriate setting and under the guidance of a healthcare professional. Like with most prescription medications, exceeding the recommended dosage or taking the medication more frequently than prescribed can cause side effects or health complications that wouldn't usually occur when used as prescribed.
A 2002 review discussed one fatal instance of a Viagra overdose where autopsy results showed a medication concentration in the blood over four times the recommended level. So, while dependency and overdose aren't really possible when used as prescribed, there can be very severe consequences from overuse.