You can use a regular expression with a look-behind assertion, which has the form (?<=...). That means "look for text that follows what is in the place of the three dots".
library(stringr)
#> Warning: package 'stringr' was built under R version 3.5.3
strings <- c("/run/media/bb/cc/GA/DrRao/JOBS/Edgar filings_full text/Form S-1/10/10_S-1_2013-11-20_0001104659-13-086087.txt", "/run/media/bb/cc/GA/DrRao/JOBS/Edgar filings_full text/Form S-1/1001172/1001172_S-1_2013-01-20_0001104659-13-086087.txt")
numbers <- str_extract(strings,"(?<=S-1/)\\d+")
numbers
#> [1] "10" "1001172"
Created on 2019-10-28 by the reprex package (v0.3.0.9000)