  We report an unusual nearly ferromagnetic, heavy-mass state with a
surprisingly large Wilson ratio $R_{\textrm{w}}$ (e.g., $R_{\textrm{w}}\sim$
700 for $x =$ 0.2) in double layered ruthenates
(Sr$_{1-x}$Ca$_{x}$)$_{3}$Ru$_{2}$O$_{7}$ with 0.08 $< x <$ 0.4. This state
does not evolve into a long-range ferromagnetically ordered state despite
considerably strong ferromagnetic correlations, but freezes into a
cluster-spin-glass at low temperatures. In addition, evidence of non-Fermi
liquid behavior is observed as the spin freezing temperature of the
cluster-spin-glass approaches zero near $x \approx$ 0.1. We discuss the origin
of this unique magnetic state from the Fermi surface information probed by Hall
effect measurements.
