Quantitative MRI (qMRI) acquired at the ultra-high field of 7 Tesla has been used in visualizing and analyzing subcortical structures. qMRI relies on the acquisition of multiple images with different scan settings, leading to extended scanning times. Data redundancy and prior information from the relaxometry model can be exploited by deep learning to accelerate the imaging process. We propose the quantitative Recurrent Inference Machine (qRIM), with a unified forward model for joint reconstruction and R2*-mapping from sparse data, embedded in a Recurrent Inference Machine (RIM), an iterative inverse problem-solving network. To study the dependency of the proposed extension of the unified forward model to network architecture, we implemented and compared a quantitative End-to-End Variational Network (qE2EVN). Experiments were performed with high-resolution multi-echo gradient echo data of the brain at 7T of a cohort study covering the entire adult life span. The error in reconstructed R2* from undersampled data relative to reference data significantly decreased for the unified model compared to sequential image reconstruction and parameter fitting using the RIM. With increasing acceleration factor, an increasing reduction in the reconstruction error was observed, pointing to a larger benefit for sparser data. Qualitatively, this was following an observed reduction of image blurriness in R2*-maps. In contrast, when using the U-Net as network architecture, a negative bias in R2* in selected regions of interest was observed. Compressed Sensing rendered accurate, but less precise estimates of R2*. The qE2EVN showed slightly inferior reconstruction quality compared to the qRIM but better quality than the U-Net and Compressed Sensing. Subcortical maturation over age measured by a linearly increasing interquartile range of R2* in the striatum was preserved up to an acceleration factor of 9. With the integrated prior of the unified forward model, the proposed qRIM can exploit the redundancy among repeated measurements and shared information between tasks, facilitating relaxometry in accelerated MRI.