I claim the ornamental design for a plant growing shelf system, as shown.
FIG. 1 is a perspective view of the plant growing shelf system showing my new design;
FIG. 2 is a front view thereof;
FIG. 3 is a back view thereof;
FIG. 4 is a left side plan view thereof;
FIG. 5 is a right side plan view thereof;
FIG. 6 is a top elevation view thereof; and,
FIG. 7 is a bottom elevation view thereof.
The shade lines in the figures show contour and not surface ornamentation.
The broken lines showing portions of the design in all figures are included for the purpose of illustrating environment and do not form a part of the claimed design.