First, become large. Then enlarge.
This questions addresses how (if) you can do it. Then just cast enlarge. This is in case you want to stay yourself.
Shapechange or similar
As Quadratic Wizard's answer states, you can shapechange (or true polymorph, or polymorph, or w/e) into something Huge, or something Large and then someone else casts Enlarge on you. He says Polymorph wouldn't work, but I'll state my case on how it might work (at least under one interpretation of the rules).
Monsters have special rules for grappling. The first is mentioned in the MM p. 11.
Many monsters have special attacks that allow them to
quickly grapple prey. When a monster hits with such an
attack, it doesn't need to make an additional ability check to
determine whether the grapple succeeds, unless the attack
A creature grappled by the monster can use its action to try
to escape. To do so, it must succeed on a Strength (Athletics)
or Dexterity (Acrobatics) check against the escape DC in the
monster's stat block. If no escape DC is given, assume the
DC is 10 +the monster's Strength (Athletics) modifier.
Second, creatures like the Giant Octupus can grapple, even though they have no "hands" (i.e., for them, it requires no hands). The Tarrasque itself can grapple using his jaws. TL;DR: creatures can grapple with rules different from the standard grappling from PHB - they can use their mouths, as an example.
Bite. Melee Weapon Attack: + 19 to hit, reach 10 ft., one target.
Hit: 36 (4d1 2 + 10) piercing damage. If the target is a creature,
it is grappled (escape DC 20). [...]
That said, let's check some beasts that can grapple. As already mentioned, the Giant Octopus can do it.
Tentacles. Melee Weapon Attack: +5 to hit, reach 15 ft., one
target. Hit: 10 (2d6 + 3) bludgeoning damage. If the target is a
creature, it is grappled (escape DC 16).
Note that it doesn't even state a Size restriction. Even if you want to consider it, Giant Octopus is Large and can be Enlarged to Huge, following the general rule as well as the specific rules.
If you really want to say that tentacles are hands, I'll use my mouth. Giant Toad is again a Large Beast that can grapple without using hands.
Bite. Melee Weapon Attack: +4 to hit, reach 5 ft., one target.
Hit: 7 (1d10 + 2) piercing damage plus 5 (1d10) poison
damage, and the target is grappled (escape DC 13).
Anyway, there are a few creatures that can grapple without using hands, and you can grapple with all of them. Other spells also don't have this limitation, so it's fine.
Well, it wouldn't be a complete answer if "Wish" wasn't here, right?