Large-scale convolutional neural networks (CNNs) suffer from very long training times, spanning from hours to weeks, limiting the productivity and experimentation of deep learning practitioners. As networks grow in size and complexity, training time can be reduced through low-precision data representations and computations. However, in doing so the final accuracy suffers due to the problem of vanishing gradients. Existing state-of-the-art methods combat this issue by means of a mixed-precision approach utilising two different precision levels, FP32 (32-bit floating-point) and FP16/FP8 (16-/8-bit floating-point), leveraging the hardware support of recent GPU architectures for FP16 operations to obtain performance gains. This work pushes the boundary of quantised training by employing a multilevel optimisation approach that utilises multiple precisions including low-precision fixed-point representations. The novel training strategy, MuPPET, combines the use of multiple number representation regimes together with a precision-switching mechanism that decides at run time the transition point between precision regimes. Overall, the proposed strategy tailors the training process to the hardware-level capabilities of the target hardware architecture and yields improvements in training time and energy efficiency compared to state-of-the-art approaches. Applying MuPPET on the training of AlexNet, ResNet18 and GoogLeNet on ImageNet (ILSVRC12) and targeting an NVIDIA Turing GPU, MuPPET achieves the same accuracy as standard full-precision training with training-time speedup of up to 1.84× and an average speedup of 1.58× across the networks.