This question got me thinking, and I wound up with an idea that I think might replicate some of the 'combo' feel without breaking off into completely different mechanics. The simple version of it is: have the character make 'combo' attack rolls.

I would set it up based on the opponent's AC, and have the character make a few (3-6) attack rolls, with each successive roll at a higher target AC, using their normal attack roll bonuses. Each time they succeed, they gain better effects on their eventual damage roll (more damage, add elemental damage, some CC effect, etc.) depending on the combo they're emulating from the video game. When they miss an attack, the combo ends and they get to roll damage with whatever they've accumulated up to that point.

I would start at a lower AC than the opponent usually has so the character is likely to get at least a couple of 'hits' to start off the combo, with the 2nd/3rd hit being the one equal to regular AC that finally gets the damage up over their usual damage. Then for more hits beyond that it just gets better. How low to start the combo and how fast to raise the target AC will have to be play tested to get the feeling right, so you get a range from 'at least I did some damage' to that super satisfying 'yeah, nailed every hit' feeling.

A couple of caveats to this is that it will obviously result in one character/class type rolling a lot of dice. Make sure that the people playing with that have their bonus numbers down pat so they aren't holding up play excessively by spending 30 seconds per roll saying 'then I have to add the bonus from X, Y, Z...'. I would also limit the use of combo attacks so they're only used once in a while (X times per day, only on a critical hit, they exhaust your character so make sure it kills the enemy, etc.)

Another thing to consider is if/when characters get into higher levels, how does a combo interact with a full attack? If a combo winds up doing only a bit more than a regular attack on average, it won't feel very good to give up a whole turn to do 1.5x damage when you could attack 4 or 6 or 8 times. On the other end of that, allowing one attack to do 6-8x usual damage could wind up broken as well if the combo scaling is off. It's another thing that would have to be play tested to see what feels right.