Magnetic multilayer devices that exploit magnetoresistance are the backbone of magnetic sensing and data storage technologies. Here, we report multiple-spin-filter magnetic tunnel junctions (sf-MTJs) based on van der Waals (vdW) heterostructures in which atomically thin chromium triiodide (CrI3) acts as a spin-filter tunnel barrier sandwiched between graphene contacts. We demonstrate tunneling magnetoresistance that is drastically enhanced with increasing CrI3 layer thickness, reaching a record 19,000% for magnetic multilayer structures using four-layer sf-MTJs at low temperatures. Using magnetic circular dichroism measurements, we attribute these effects to the intrinsic layer-by-layer antiferromagnetic ordering of the atomically thin CrI3. Our work reveals the possibility to push magnetic information storage to the atomically thin limit and highlights CrI3 as a superlative magnetic tunnel barrier for vdW heterostructure spintronic devices.