We study the dynamic spectrum aggregation and allocation problem in cognitive radio networks (CRNs). When the available spectrum is fragmented and the bandwidth of the spectrum holes is narrow, spectrum aggregation can be a solution to use multiple holes. Since the spectrum holes can have different bandwidth and cognitive radios can dynamically adjust the centre frequency and the bandwidth for each transmission, the CRN has the high flexibility in spectrum use with aggregation. However, aggregation of fragmented spectrum can lead to channel fragmentation and further spectrum fragmentation. In this context, we formulate the dynamic spectrum aggregation problem considering practical hardware constraints whilst aiming to minimise spectrum/channel fragmentation. Then, we study the application of three well-known memory allocation techniques, to spectrum aggregation, and analyse the impacts of the spectrum aggregation on spectrum/channel fragmentation. Based on the analysis, we derive guidelines that govern the design of the efficient spectrum aggregation and allocation algorithms.