Step 1: Identify the pathway. Cells recycle purine bases through a salvage route to avoid the energetic cost of building purines from scratch. The key salvage enzyme for hypoxanthine and guanine is HGPRT.
Step 2: When HGPRT is absent, hypoxanthine and guanine cannot be re-converted to nucleotides, so they are funneled into the degradation pathway and end as excess uric acid, while PRPP accumulates and further fuels purine synthesis.
Step 3: Clinically this produces the classic triad of hyperuricemia, dystonic-choreoathetoid neurologic disease, and self-injurious biting, in an X-linked pattern affecting boys.
Step 4: Contrast the foils: a block at xanthine oxidase lowers uric acid; APRT loss precipitates adenine-derived stones; AMP deaminase loss gives exercise-induced myopathy. None of these reproduces Lesch-Nyhan, so HGPRT deficiency is the answer.
\[\boxed{\text{HGPRT deficiency}}\]