It is always satisfying to see the character you have played to grow with you, especially the characters that you haven't seen in a long time. It makes players feel involved in the life of the fictional character. You feel the emotions they feel, and you experience their growth as you grow. It actually reminds players the time they enjoyed playing the previous games and create a nostalgia atmosphere very early on.
Examples[]
Uncharted 4 A Thief's End[]
When I played Uncharted 4, I immediately had the feeling about "how time flies" the moment I saw Nathan's face. Then in the cut scene where he and Elena eat dinner at home, I remembered the adventures they both shared and the times they fought for each other. These feelings made me addicted to the game hopelessly. There are details that reminds me of their time throughout the whole game.
Telltale's The Walking Dead[]
Telltale is dedicated to make sure that the decisions you take in their game have meaningful consequences. In their Walking Dead series of games, the decisions the player takes has a direct affect on the characters' emotional growth. You can even carry over your save between the sequels to see how your previous decisions affect the current game. In addition to the well developed story and character's, this ability to not only see but affect a character's growth is what makes this game special.
Mass Effect 2
Mass Effect is excellent at showing the progression of characters over the course of different games. The fi
rst ttrilogy shows characters progressing throughout all three games, changing their roles and motivations in the context of the larger universe. An excellent example is Liara T'Soni, who is a member of the player character's squad in the first game. A young, naive researcher, Shepard takes her on a whirlwind tour of the galaxy, discovering ancient tech and stopping nefarious plots. But Shepard goes missing, presumed dead at the start of the second game - and when Shepard finds their way back to Liara, the naive girl is a devious information broker, entirely unwilling to rejoin Shepard's adventures. If anything she treats Shepard coldly, scarred by events Shepard wasn't there for, and perhaps hurt by the abandonment of her commander. This nuanced response and change in relationship adds to the depth of the character, and makes her feel like a being independent of the player.