Realizing high thermal conductivity nanocomposites is a challenge because of difficulties in incorporating high fractions of uniformly dispersed nanofillers and countering low filler-matrix interfacial conductance. Here, we obviate these issues by using <3 vol. % gold nanowire fillers to obtain a 30-fold increase in polydimethylsiloxane thermal conductivity that is 6-fold higher than any nanocomposite at low nanofiller loadings and exceeds theoretical predictions. The nanowire diameter and aspect ratio are keys to obtaining cold-welded networks that enhance thermal conductivity while fostering low modulus and electrical conductivity. Such nanowire nanocomposites are attractive for many applications in electronics, packaging, and energy devices.