We analyze the scaling and cost-performance characteristics of current and projected connectomics approaches, with reference to the potential implications of recent advances in diverse contributing fields. Three generalized strategies for dense connectivity mapping at the scale of whole mammalian brains are considered: electron microscopic axon tracing, optical imaging of combinatorial molecular markers at synapses, and bulk DNA sequencing of trans-synaptically exchanged nucleic acid barcode pairs. Due to advances in parallel-beam instrumentation, whole mouse brain electron microscopic image acquisition could cost less than $100 million, with total costs presently limited by image analysis to trace axons through large image stacks. Optical microscopy at 50 to 100 nm isotropic resolution could potentially read combinatorially multiplexed molecular information from individual synapses, which could indicate the identifies of the pre-synaptic and post-synaptic cells without relying on axon tracing. An optical approach to whole mouse brain connectomics may be achievable for less than $10 million and could be enabled by emerging technologies to sequence nucleic acids in-situ in fixed tissue via fluorescent microscopy. Novel strategies relying on bulk DNA sequencing, which would extract the connectome without direct imaging of the tissue, could produce a whole mouse brain connectome for $100k to $1 million or a mouse cortical connectome for $10k to $100k. Anticipated further reductions in the cost of DNA sequencing could lead to a $1000 mouse cortical connectome.