Correct solutions came from Brian Forster, Bradley Univeristy; Dan Dima, Romania; Claudio Baiocchi, Italy; Philippe Fondanaiche, France; K. Sengupta, India; Aaron Kahn; Ahron Teitelman, Israel; Nathan Faber, USA; Stefan Gaţachiu, Romania; Lou Cairoli, USA; Farid Lian, Colombia; John Snyder, USA; Ron Welch, USA; Kathy Zhong, USA.
The children will end up with 4/5 of a cookie if the process is continue indefinitely. Here is the solution from Nathan Faber.
In the first round, n = 1, mom cuts each cookie and has three pieces let after giving 1/2 cookie to each child. For n=2, mom cuts each of these pieces, gives 1/4 cookie to each child, and has one piece left. For n=3, mom cute this piece into eight, gives 1/32 cookie to each child, and has three pieces left. If she continues this process, she will have one piece left whenever n is even and three pieces left whenever n is odd. If she has one piece, she must divide it into eight new pieces, and if she has three pieces, she has to make one division. Therefore, the sequence of cookie fragments that each child receives is 1/2, 1/4, 1/32, 1/64, 1/512, 1/1024, 1/8192, 1/16384,... which is obtained by alternately multiplying the terms by 1/2 and 1/8. The sum of the terms in pairs is 3/4, 3/64, 3/1024,...,3/4·3/16n,... This is a geometric sequence whose sum is 4/5.