Wikipedia has an article on computer addiction, a fairly broad term that seems to include the situation you are talking about. It begins like this:
Computer addiction is a mental illness which causes the excessive use of computers to the extent that it interferes with daily life. Excessive use may explain problems in social interaction, mood, personality, work ethic, relationships, thought processes, or sleep deprivation. The Diagnostic and Statistical Manual of Mental Disorders does not include a diagnosis for such a disease.
The term computer addiction originated long before the Internet.
Some people develop bad habits in their computer use that cause them significant problems in their lives. The types of behavior and negative consequences are similar to those of known addictive disorders.
This is connected with the related Internet addiction disorder, or IAD:
Internet addiction disorder (IAD), or, more broadly, Internet overuse, problematic computer use, or pathological computer use, is excessive computer use that interferes with daily life. These terms avoid the term addiction and are not limited to any single cause.
That one may be more appropriate, since it zeroes in on the networking aspect more than on the gadget aspect, which I believe is what you were talking about. The term is not without controversy, and does not appear in the DSM-IV. The same article later says:
Online activities which, if done in person, would normally be considered troublesome, such as compulsive gambling, or shopping, are sometimes called net compulsions. Other habits such as reading, playing computer games, or watching a staggering amount of internet videos or movies are all troubling only to the extent that these activities interfere with normal life. Supporters of disorder classification often divide IAD into subtypes by activity, such as excessive, overwhelming, or inappropriate pornography use, gaming, online social networking, blogging, email, or Internet shopping. Opponents note that compulsive behaviors may not themselves be addictive.
Over the past decade, the concept of Internet addiction has grown in terms of acceptance as a legitimate clinical disorder often requiring treatment. However, known academic authorities take stances in either supporting or opposing the existence of Internet addiction disorder (IAD). In 2006, the American Medical Association declined to recommend to the American Psychiatric Association that they include IAD as a formal diagnosis in DSM-V, and recommended further study of "video game overuse."
Whether to call all this an actual “addiction” — or not — may matter only to one’s health insurance provider. If it were in the DSM, then insurance companies would likely have to pay for addiction “treatments”.