To reduce the network load during peak hours, servers deliver partial data to users during the off-peak time of the network before the actual requests are known, which is known as caching. This paper studies a single user caching problem in which the file contents are subject to dynamic modifications with respect to a certain probability distribution. To cope with the dynamical nature of the file contents, a successive refinement approach to caching is presented: partial information of the original data is cached first and then if there is a modification, a refinement to the previously cached data is delivered to the user. Given a fixed cache memory, there is a tension between the rates of two cache descriptions. The problem of optimal caching strategies is formulated through a successive Gray-Wyner network, the optimal rate region of which is characterized. Some lower and upper bounds on the performance of optimal caching strategies are developed and shown to actually yield closed form solutions for certain classes of file contents.