While there is no proof that this is the smallest number, you can find an rough upper bound by shedding digits After a couple minutes of playing I worked it do

While there is no proof that this is the smallest number, you can find an rough upper bound by shedding digits After a couple minutes of playing I worked it down to 210000041821 before getting bored. A tenth of this and you are in telephone number space. Numbers a bit smaller than this and you are in valid parcel tracking numbers. Either Google indexing this page or more packages being sent will eventually invalidate this post.