Is there a way to extract caffeine from coffee completely?
Yes, this is how decaffeinated coffee is made. If I recall correctly, this is done commercially by dissolving the caffeine in dichloromethane (DCM), or more recently in supercritical carbon dioxide. This switch is probably over concerns regarding chronic industrial exposure to DCM, which is mildly toxic.
From personal experience, I have extracted caffeine from instant coffee using a separatory funnel with a system of water and dichloromethane. Purification by sublimation finished the job.
The short answer is no.
But for all intents and purposes, yes.
By submitting coffee to several small volume extractions, one can drastically reduce the concentration of caffeine present in the grains. This method of extraction (multiple small volume aliquots) is more effective than a single large volume extraction. The link I shared provides a good discussion.