You can get a lot of mileage out of them, especially (though not always) when dealing with numbers not particularly close to a round figure. Not terribly useful with 13*18, except to identify 13 as a prime that cannot BE factored, but 13*9*2 is not too bad (though 180+54 or even 20*13-26 are both better.) For 72*44, 2^3*3^2*2^2*11 works pretty well, because then it is just 99*32=3200-32=3168. Beyond a point, as you note, it just becomes too big a pain to remember one set of numbers accurately while mentally playing with a second; people good at blindfolded chess probably do it quite well.

To do 1377*8513 my own approach would be 1400*8500=11,900,000; subtract 23*8500 (195,000)=11,704,500; add 13*1377 (17,901)=11,722,401.

I will not pretend I did all that mentally; I gave it the old college try while outside smoking, but once I got to 11,704,500 I 1) forgot to count the 1000s in 13*1377 and 2) far worse, SUBTRACTED rather than adding the result, so I wound up with 11,700,369. Generally, if I have to track more than 5 or 6 digits while doing a separate operation, it is time to write a few of them down before I forget one.