The connectedness percolation threshold (phi(c)) for spherically symmetric, randomly distributed fractal aggregates is investigated as a function of the fractal dimension (d(F)) of the aggregates through a mean-field approach. A pair of aggregates (each of radius R) are considered to be connected if a pair of primary particles (each of diameter delta), one from each assembly, are located within a prescribed distance of each other. An estimate for the number of such contacts between primary particles for a pair of aggregates is combined with a mapping onto the model for fully penetrable spheres to calculate phi(c). For sufficiently large aggregates, our analysis reveals the existence of two regimes for the dependence of fc upon R/delta namely: (i) when d(F) > 1.5 aggregates form contacts near to tangency, and phi(c) approximate to (R/delta)(dF-3), whereas (ii) when d(F) < 1.5 deeper interpenetration of the aggregates is required to achieve contact formation, and phi(c) approximate to (R/delta)(-dF). For a fixed (large) value of R/delta, a minimum for fc as a function of dF occurs when d(F) = 1.5. Taken together, these dependencies consistently describe behaviors observed over the domain 1 <= d(F) <= 3, ranging from compact spheres to rigid rod-like particles.