Modern data centres provide large aggregate capacity in the backbone of networks so that servers can theoretically communicate with each other at their maximum rates. However, the Transport Control Protocol (TCP) cannot efficiently use this large capacity even if Equal-Cost Multi-Path (ECMP) routing is enabled to exploit the existence of parallel paths. MultiPath TCP (MPTCP) can effectively use the network resources of such topologies by performing fast distributed load balancing. MPTCP is an appealing technique for data centres that are very dynamic in nature. However, it is ill-suited for handling short flows since it increases their flow completion time. To mitigate these problems, we propose Maximum MultiPath TCP (MMPTCP), a novel transport protocol for modern data centres. Unlike MPTCP, it provides high performance for all network flows. It also decreases the bursty nature of data centres, which is essentially rooted in traffic patterns of short flows. MMPTCP achieves these nice features by randomising a flow's packets via all parallel paths to a destination during the initial phase of data transmission until a certain amount of data is delivered. It then switches to MPTCP with several subflows in which data transmission is governed by MPTCP congestion control. In this way, short flows are delivered very fast via the initial phase only, and long flows are delivered by MPTCP with several subflows. We evaluate MMPTCP in a FatTree topology under various network conditions. We found that MMPTCP decreases the loss rate of all the links throughout the network and helps competing flows to achieve a better performance. Unlike MPTCP with a fixed number of subflows, MMPTCP offers high burst tolerance and low-latency for short flows while it maintains high overall network utilisation. MMPTCP is incrementally deployable in existing data centres because it does not require any modification to the network and application layers.