We examined the growth trajectories of reading in a consistent orthography (Greek) in two developmental periods (from Grade 1 to Grade 4 and from Grade 4 to Grade 10) and what cognitive skills predict the growth patterns. Seventy-five Greek-speaking children were assessed in Grades 1, 2, 4, 6, and 10 on word-, nonword-, and text-reading fluency. In Grades 1 and 4, they were also assessed on phonological awareness, rapid naming, phonological memory, orthographic knowledge, and articulation rate. Results of growth curve modeling showed that during the first developmental period, there was a rapid initial growth from Grade 1 to Grade 2 followed by a less rapid growth from Grade 2 to Grade 4. In the second developmental period, the slow growth continued. In both developmental periods, rapid naming and orthographic knowledge predicted the initial status of all reading outcomes and phonological memory predicted the initial status of nonword-reading fluency. Phonological awareness predicted the initial status of nonword-reading fluency in the first developmental period and the initial status of word- and text-reading fluency in the second developmental period. None of the cognitive skills predicted the growth rate in reading skills. Theoretical and practical implications of these findings are discussed.