Cloud services and other shared third-party infrastructures allow individual content providers to easily scale their services based on current resource demands. In this paper, we consider an individual content provider that wants to minimize its delivery costs under the assumptions that the storage and bandwidth resources it requires are elastic, the content provider only pays for the resources that it consumes, and costs are proportional to the resource usage. Within this context, we (i) derive worst-case bounds for the optimal cost and competitive cost ratios of different classes of "cache on $M^{th}$ request" cache insertion policies, (ii) derive explicit average cost expressions and bounds under arbitrary inter-request distributions, (iii) derive explicit average cost expressions and bounds for short-tailed (deterministic, Erlang, and exponential) and heavy-tailed (Pareto) inter-request distributions, and (iv) present numeric and trace-based evaluations that reveal insights into the relative cost performance of the policies. Our results show that a window-based "cache on $2^{nd}$ request" policy using a single threshold optimized to minimize worst-case costs provides good average performance across the different distributions and the full parameter ranges of each considered distribution, making it an attractive choice for a wide range of practical conditions where request rates of individual file objects typically are not known and can change quickly.