Carin Baer/AMC
For a long-running TV show, child actors are almost always a headache. Putting aside all the ethical ramifications of thrusting someone into the public eye at such a young age, there's the simple fact that kids grow and change in ways that often contradict what the series itself is going for. Sometimes the actor hits puberty and starts getting too self-conscious, or sometimes they simply hit puberty too soon (or too late) and that changes the sort of storylines the writers can do with them. ("Lost," a show whose first three seasons only span three months, particularly struggled in this regard when it came to the character of Walt.)
"Mad Men" lucked out: it only had two main child actors to deal with, and they struck gold with one of them. Kiernan Shipka, who plays Sally Draper, sticks around for the whole series, getting plenty of her own storylines to work with. But with the actors who played Bobby Draper, who's about three years younger than Sally, it was a different story. The first Bobby was played by Maxwell Huckabee for most of season 1. In a 2015 interview with Entertainment Weekly, Shipka said that the Bobby #1 actor "holds a special place in my heart."
Bobby #2 was played by Aaron Hart, and he stuck around for the tail end of season 1 and all of season 2. The Draper son was still mostly a background character during this period of the show, although he did get that one standout scene where he talks with Don about Don's father. Shipka also remembered Hart fondly, describing him as having "had some pretty cool boots."
In season 3, Hart was replaced by Bobby #3, Jared Gilmore, who left after season 4 because he was offered a bigger, more promising role in the ABC show "Once Upon a Time." This was the point in "Mad Men" where the disparity between Bobby and Sally became clear; Sally was getting fleshed-out storylines, including plenty of scenes without Betty or Don in the room with her, whereas Bobby stayed in the background. It was around here that fans wondered if any of the adult characters cared about Bobby, because it sure seemed like Sally was getting all the attention. (Even AMC considered making a "Mad Men" spinoff centered on Sally at one point.)
Bobby #4: the reigning champ amongst the Bobbies
AMC
Season 5 introduced Mason Vale Cotton, who stuck around to play Bobby until the series finale. This makes him the fourth official Bobby, although Shipka has mentioned that there were actually four other Bobby actors we never got to see. One of them showed up briefly in the pilot, which was filmed a year before the rest of season 1. (This is also the case for Sally; "I was actually Sally #2," Shipka joked.) The other three stepped in to play Bobby during brief scenes where his face was covered — Shipka called two of them Bucket Head Bobby and Pillow Face Bobby. Shipka had nothing but kind words to say about all the Bobbies, but by the nature of their limited time together she was never that close to them.
While we're not sure if the average viewer noticed all the different Bobby Drapers throughout the show, it's clear that the online fandom has been well aware. When "Mad Men" fans talk about the character, they'll often throw a number designation to make clear which version of Bobby they're referring to. There's also an ongoing fan debate over whether Bobby's lack of screen time is an intentional commentary on Don and Betty's bad parenting or if it's the result of the show's writers not quite knowing what do with the kid.
Either way, things improved for Bobby once the series settled on a long-term actor for him in season 5. Sure, Cotton didn't get a ton to do in the role in season 5, but by season 6 the writers started trusting him to pull off scenes with substance. There was that sweet storyline where Bobby and Don go see "Planet of the Apes" together, and the darker storyline where Don and Betty reconnect in "The Better Half" while Bobby's constantly trying to hold their attention. Maybe most memorable is Bobby's brief season 7 subplot where he gives Betty's sandwich to a girl he likes on the field trip, causing Betty to give him the silent treatment for the rest of the day.
In addition to getting more juicy material, Bobby was also handed some strong comedic relief moments, like his "Are we negroes?" reaction to a home invasion in "The Crash" (aka one of the best episodes of "Mad Men") or his surprisingly mean outburst when Betty gets her hair dyed. Sure, he's still overshadowed by Sally, but Bobby #4 was a clear step up from Bobbies #1-3. Bobby never escaped his Forgotten Middle Child syndrome, but at least the show itself didn't take Bobby #4 for granted.