Stardew Valley allows the player to have a maximum of two kids at one time. As mentioned above, you can use the Dark Shrine of Selfishness to dismiss your children and turn them into doves, leaving your home empty once again. Then you will gain the opportunity to have new kids. So, in theory, you can have as many kids as you like in Stardew Valley, but you'll have to abandon your old ones to get new ones.