Other than delving into the Fighter class for 4 attacks per action, I have found very few ways to expand your repertoire of attacks per turn to put towards a brawler build that emphasizes shoving and grabbing combatants to control the field. I had a few suggestions, Haste and the Tavern Brawler feat, commented at the time, but I'm wondering what else might be there?
There are a few different ways to go about this:
Like you say, a fighter's Extra Attack lets you make up to four attacks per turn, but many classes will let you make two attacks - Barbarian, Monk, Paladin and a bunch of subclasses, which cover most brawler builds.
Haste is one of the few effects that gives you a second action on a turn, and is thus a great way to get an extra grapple attempt in. Of course, you do need five levels in a spellcasting class to get this, as opposed to the other way to get an extra action.
Twenty levels of fighter for four attacks is a bit much, but a two-level dip gets you this beauty. Only once per short rest, which is a bit of a bummer, but on the other hand you can use Extra Attack with it.
Tavern Brawler and Shield Master
I list these together as they both use your bonus action, but these feats (which are an optional rule, but a popular one) do provide different options - Tavern Brawler lets you grapple as a bonus action when you hit with an unarmed or improvised strike and Shield Master lets you shove when you take the Attack action while holding a shield.
Thunderwave, Gust of Wind, Entangle...
As a slightly left-field suggestion (though compatible with getting Haste), a lot of spells let you push people around or hold them in place. Thunderwave pushes a group of people back from you, Gust of Wind consistently pushes in a line, and Entangle effectively grapples a whole bunch of people. It doesn't let you make more attempts per se, but a single casting of one of these spells is equivalent to several push or grapple attempts.
In addition to @BlueHairedMeerkat's answer:
The monk's Way of the Open Hand monastic tradition grants the Open Hand Technique feature, which allows you to shove an opponent with your attack (requires a ki point to use Flurry of Blows):
Whenever you hit a creature with one of the attacks granted by your Flurry of Blows, you can impose one of the following effects on that target:
It must succeed on a Dexterity saving throw or be knocked prone.
It must make a Strength saving throw. If it fails, you can push it up to 15 feet away from you.
The Sentinel feat allows you to stop an opponent's movement (not exactly grappling, but controlling the field in the same way):
When you hit a creature with an opportunity attack, the creature's speed becomes 0 for the rest of the turn.
Advantage essentially functions as a new chances to grapple since you roll a second d20 so a fail can be ignored. Grapples and Shoves are not attack rolls so getting advantage is different than it would be for regular attacks. Below are a few options:
- The enhance ability spell for Bull's Strength
- The Help action (such as from a familiar)
- Barbarian feature: Rage