Iterated-Logarithmic Size of Gross Exceptional Sets

For each q>1, we construct a transcendental entire function and one regular inverse germ whose exceptional directions in Gross's star theorem contain a compact set of positive finite Hausdorff measure for h_q(t)=t[log log(e^e/t)]^q. The same inverse branch escapes at unit radius in every chosen direction. This single example has infinite measure for every gauge t[log(e/t)]^epsilon, epsilon>0, while remaining of length zero. The six-page proof refines Sina Nadi's positive-capacity construction using slowly varying corridors, a nonatomic Bernoulli shell sum and an exponential holomorphic approximation weight. Nadi's capacity result and Eremenko's public dimension-one attribution are credited; neither is claimed as new here. The endpoint q=1, equality with the entire exceptional set and a full exceptional-set classification are excluded. The package includes 16,878 supporting exact/symbolic and high-precision regression checks; these do not replace the written analytic proof. Developed with AI assistance and self-audited, this is an unrefereed preprint without independent review, formal verification or certified absolute priority.