Step 1: Recall the building blocks of H. pylori eradication. A typical course combines an acid suppressant with antibiotics such as amoxicillin, clarithromycin, and metronidazole, sometimes with tetracycline and bismuth in quadruple therapy.
Step 2: Each of clarithromycin, metronidazole, and amoxicillin appears in recognized triple or quadruple regimens, so all three are legitimate treatments.
Step 3: Gentamycin is an aminoglycoside used chiefly for serious gram-negative systemic infections; it does not feature in any validated anti-H. pylori protocol because it is poorly suited to the gastric niche.
Step 4: Since the question seeks the drug not used, gentamycin is the odd one out.
\[\boxed{\text{Gentamycin}}\]