For reference, I built a home PC that I successfully do deep learning and data analysis on (mostly tensorflow and scipy stack) for about ~$10k. It's liquid cooled, has 15 fans, four radiators, an i7-6900K CPU, 128GB RAM, four GTX 1080 GPUs (controversial), four TBs of HDD space and 1TB of SSD space. I don't recommend you start with this at all, but my point is that porting your hardware from point A to point B will be a pain if it comes to it.
I used the guide here as a reference about 8 months ago when I built it: http://graphific.github.io/posts/building-a-deep-learning-dr.... My purpose in doing this was, essentially, to pay for electricity rather than AWS/GCP/Azure compute resources (and in that regard it's been very successful!).
I know I'm hijacking a thread here to talk about building home machines for professional deep learning work when this story is clearly not intended for that, but I wanted to throw in this perspective so that it's understood this is very different from just "build this machine to start out and upgrade it later." There's a law of diminishing returns here, but in general my point is that I do not think this is a minimum for "start doing deep learning effectively at home." If you want to learn hands on deep learning cheaply, my opinion is that it would be more efficient to use compute resources from a cloud provider before diving into this with a home-based custom machine.
tl;dr: The demographic of folks who probably want/should/need to build a home deep learning machine probably has little overlap with the demographic of folks who want to do it non-professionally, or at least with only $1k in resources.