No, it's not.
When lifting the other character you are either taking an action, or you interacting with an object -- depending on your DM's rulings.
If the DM rules that the pile driver is an attack (which I'd argue it is), that would cost you an action, and you can only repeat it if you have movement and an action to spend on it. This assumes you've grappled them, and have an attack left.
If the DM rules you are use an object. If it wasn't an enemy, you'd only be allowed to pick it up once. Once you drop it (which is free), it is no longer a free action to pick it up again. Use An Object action:
You normally interact with an object while doing something else, such
as when you draw a sword as part of an attack. When an object requires
your action for its use, you take the Use an Object action. This
action is also useful when you want to interact with more than one
object on your turn. (PHB 193)
Since you can't lift someone and do something else, it might take a full action to lift someone above your head. Remember that a round is 6 seconds, dropping someone more than once (or twice if you have the actions for it) would take about that long. Moreover, really can't lift someone and jump while doing something else, so it could be argued it can't be an interaction and must be a "Use Object" action.
To rule otherwise would allow a free 5d6 damage without using any action, and that, while awesome the first time (and a DM may allow it once, by Rule of Cool), is game breaking.
By rules as written, you can only grapple creatures one size larger than yourself (PHB p. 195), so a medium creature can jump on a large and a small onto a medium.
Note that the second paragraph is an alternative, not a variant, so it too is in play. If the creature is larger than you can grapple and is suitably large, whatever the DM decides that is, you can simply use your movement to walk or climb it. The DM would need to decide which and if the creature was difficult terrain or not and if any checks are needed; presumably Dexterity (Acrobatics). Assuming the creature didn't want you there it could try to throw/pull/buck you off, resolve this as an opposed check.
Personally, I think it would be awesome if the players jumped and climbed up the dragon to try and sever its spine. They would too; right up to the point where I say "The dragon drops prone and rolls over you. Time for some saving throws."
Best Answer
You've misinterpreted the movement penalty for grappling and the movement restrictions for jumping. They limit different things.
From the PHB, page 195:
So you're not halving your jump distance, you're halving your movement speed. And you have to use that for jumping (PHB page 182):
This means that, for a player whose normal movement speed is 30ft, while grappling he can move 15ft, and that's also the maximum jump height he can achieve. Not even the Jump spell or the Boots of Striding and Springing will let him jump further than that. The effects, however, do let him get around the Standing High Jump penalty.
To summarize, when looking into how high you can possibly jump while grappling, you make the calculations in a few steps:
You can use this little tool I whipped up to test out various combinations.
If you find the maximum jump height (step 2) to be restricting you more than your jump distance achievable (step 4), you can use spells and items that increase your movement speed to propel you further. The Haste spell is good to consider. While it is a substantially higher level spell, it lets you use an extra action to Dash, quadrupling your speed in total. The added AC is a nice touch, too. You can even combine it with the Jump spell, as the latter does not require Concentration.
Using just the effects in your example, however, the highest you can jump is 15ft, using a Standing High Jump. If you somehow throw in a Dash, you can jump twice as much. Note that, if the PC's base Strength is already 18, the Belt does not help increase the jump height in this case.
Keep in mind, though, one litle bit about the High Jump rules:
The rule enables you as a DM to let your player jump beyond his maximum movement speed (this bit of the rule is written directly after the bit about the movement speed restriction). When can that happen, how much of a bonus does the player get, what the DC is and what the possible consequences are (chance of Exhaustion, for example) - all of this is up to you.
A commentary on the Belt of Storm Giant Strength
I personally dislike the use of this item. First of all, it is a legendary item, suitable only for the most glorious of adventurers. Giving this item to a player character that's not very high in level (18 at least, I'd say) would seem overpowered, and frankly uninteresting. Even at that level it might not be as good as it sounds. Surely, it's a fantastic item, but it makes all the precious Ability points a grappler had spent on Strength a waste. On the other hand, having the belt in possession early and spending the points somewhere else (or getting feats), the grappler's main mechanic would now depend entirely on the belt. Losing the belt would thus make the character almost useless. While you might not plan on having it taken away from him in any encounter, the player might become paranoid about it nonetheless.
Alternative to using the belt, using a Haste spell (3rd lvl) and the Boots of Striding and Springing (uncommon item), the player can reach 21ft with a Strength of 18, and 24ft with a strength of 20 (assuming a base movement speed of 30ft, and making a Dash with the Haste extra action) by making a normal High jump (with a 10ft running start). Throw in a Jump spell (1st lvl) and the High jump goes up to 50ft. Using these three effects, the Belt would only give an improvement of 4ft with a Standing high jump.
If the Haste spell is not available, and using Dash is not an option, then a better combo would be to use Boots of Speed (rare item) and the Jump spell. Just those two bring the High jump up to 20ft, which might be enough. In this case, too, increasing Strength above 18 brings no benefits (though I'm sure it does so in the other aspects of grappling).
Seeing as all the Belts of Giant Strength do provide great bonuses to almost all aspects of grappling, I would award the legendary version at lvl 20 as a special Grappler capstone. Of course, not without effort. It's a nice excuse to lead your players into a perilous dungeon. I would also house-rule that the belt require a Strength of 20 to wear, just so all those Ability points don't feel like they're going to waste. If your player left his strength at 18, though, that might not be necessary.
At lower levels, a potion of Giant Strength (whichever tier you fancy) might be a suitable reward for a heroic deed, and since it's only temporary it wouldn't break anything (including fun).