Previously in Magic, "infinite" meant an arbitrarily large number. Infinity Elemental changes that. It allows for infinite power and infinite damage on its own, but what else can we get infinite of using combinations of other cards? (These don't have to be practical or useful in any way. They're just for the sake of seeing what's possible.)
Remember, you aren't allowed to repeat a loop an infinite number of times. So, you can't use infinite of a resource to pay for the same ability over and over. (so no using infinite life to draw cards with Griselbrand, for example.) You can activate infinite separate abilities, if you create infinite separate permanents with the same ability. You can have an infinite number of triggers created by infinite of something else.
Obviously you can get infinite life with Lifelink. Note that this doesn't allow us to get infinite of something by paying life, because that's still a repeatable but finite loop. We can still use infinite life to get:
Using Infinity Elemental's power with other cards, we can also:
Deal infinite damage to a creature using any fight spell.
Get infinite life loss from Dread Defiler.
Create a creature with base power AND toughness of infinity using Kalitas, Bloodchief of Ghet.
Get infinite energy with Peema Aether-Seer.
Draw infinite cards with Prime Speaker Zegana.
Discard infinite cards with Tormented Thoughts.
Use Tainted Strike to put infinite -1/-1 counters on a creature. (The only way to make them stick around is with a creature that has infinite toughness.)
Get infinite +1/+1 counters on any creature with Vigor.
Give something - infinity / - infinity with Vish Kal, Blood Arbiter
Of course, that's not all. With infinite tokens from the previously listed sources, we can then:
Get infinite colored mana with Cryptolith Rite
Use Aura Shards to destroy an infinite number of artifacts or enchantments.
Use Emrakul's Evangel to sacrifice an infinite number of creatures.
I'm sure there's a lot more possibilities we can find here - I haven't even looked at what you can do with Mycosynth Lattice or things that rigger when creatures die. So what else can we make infinite of?
While you are right that Infinity Elemental opens up a whole lot of sillyness, some of the cards you mentioned don't work the way you think they do.
For starts, you mentioned the wrong Ajani if you want infinite tokens from infinite life. You want Ajani, Caller of the Pride, not Ajani Goldmane. Kalitas, Bloodchief of Ghet makes a token with the same power as the destroyed one. And the same toughness of the destroyed one. So if he destroys Infinity Elemental, you don't get an infinite/infinite token, you get a infinite/5 token. Energy Tap adds mana equal to the tapped creatures CMC, not it's power. So no infinite mana there.
Also probably not a good idea to draw infinite cards with Prime Speeker, Zegana as your deck doesn't have infinite cards in it. So you would just lose.
Step 1: Abyssal Persecutor and Grusilda, Monster Masher on the battlefield, Infinity Elemental and Earl of Squirrel in the graveyard.
Step 2: Grusilda reanimates an Infinite Squirrel Monstrosity.
Step 3: Attack with the Infinity Squirrel, generating an infinite number of Squirrel tokens.
Step 4: Chitter menacingly.
Have Echo infinity with Volcano Hellion (not particularly useful but it's something we haven't mentioned).
If we create infinity creatures (assuming we cast Mercy Killing on Infinity Elemental) and then cast Wrath of God then that gives us infinity dies/leaves the battlefield triggers (which also means infinity targets). That means:
Infinity charge counters on Black Market and infinity Eyeball counters on Jar of Eyeballs
Infinite untap triggers with Thornbite Staff, Galvanic Juggernaut or Goblin Sharpshooter (assuming we make them indestructible)
Infinite tap triggers for Fleshmad Steed
We can use Bitter Ordeal to have infinity sorcery spells on the stack, and Infinity library shuffles (for Cosi's Trickster I guess?
Infinity experience counters from Meren of Clan Nel Toth (which leads to infinity P/T creatures for Daxos and Kalemne, instants and sorceries costing infinity less and ezuri Putting infinity +1/+1 counters on.)
Also probably not a good idea to draw infinite cards with Prime Speeker, Zegana as your deck doesn't have infinite cards in it. So you would just lose.
Remember, you aren't allowed to repeat a loop an infinite number of times. So, you can't use infinite of a resource to pay for the same ability over and over. (so no using infinite life to draw cards with Griselbrand, for example.) You can activate infinite separate abilities, if you create infinite separate permanents with the same ability. You can have an infinite number of triggers created by infinite of something else.
Obviously you can get infinite life with Lifelink. Note that this doesn't allow us to get infinite of something by paying life, because that's still a repeatable but finite loop. We can still use infinite life to get:
Infinite tokensAjani, Caller of the Pride.
Infinite toughness from Serra Avatar.
Using Infinity Elemental's power with other cards, we can also:
Deal infinite damage to a creature using any fight spell.
Get infinite life loss from Dread Defiler.
Create a creature with base power AND toughness of infinity using Kalitas, Bloodchief of Ghet.
Get infinite energy with Peema Aether-Seer.
Draw infinite cards with Prime Speaker Zegana.
Discard infinite cards with Tormented Thoughts.
Use Tainted Strike to put infinite -1/-1 counters on a creature. (The only way to make them stick around is with a creature that has infinite toughness.)
Get infinite +1/+1 counters on any creature with Vigor.
Give something - infinity / - infinity with Vish Kal, Blood Arbiter
Of course, that's not all. With infinite tokens from the previously listed sources, we can then:
Get infinite colored mana with Cryptolith Rite
Use Aura Shards to destroy an infinite number of artifacts or enchantments.
Use Emrakul's Evangel to sacrifice an infinite number of creatures.
I'm sure there's a lot more possibilities we can find here - I haven't even looked at what you can do with Mycosynth Lattice or things that rigger when creatures die. So what else can we make infinite of?
For starts, you mentioned the wrong Ajani if you want infinite tokens from infinite life. You want Ajani, Caller of the Pride, not Ajani Goldmane.
Kalitas, Bloodchief of Ghet makes a token with the same power as the destroyed one. And the same toughness of the destroyed one. So if he destroys Infinity Elemental, you don't get an infinite/infinite token, you get a infinite/5 token.
Energy Tap adds mana equal to the tapped creatures CMC, not it's power. So no infinite mana there.
Also probably not a good idea to draw infinite cards with Prime Speeker, Zegana as your deck doesn't have infinite cards in it. So you would just lose.
Step 2: Grusilda reanimates an Infinite Squirrel Monstrosity.
Step 3: Attack with the Infinity Squirrel, generating an infinite number of Squirrel tokens.
Step 4: Chitter menacingly.
Soul Conduit or Arbiter of Knollridge (and possibly Reverse the Sands) can give your opponents infinity life too. Not sure how useful that is but it has to open some doors, no?
Have Echo infinity with Volcano Hellion (not particularly useful but it's something we haven't mentioned).
If we create infinity creatures (assuming we cast Mercy Killing on Infinity Elemental) and then cast Wrath of God then that gives us infinity dies/leaves the battlefield triggers (which also means infinity targets). That means:
Infinity charge counters on Black Market and infinity Eyeball counters on Jar of Eyeballs
Infinite untap triggers with Thornbite Staff, Galvanic Juggernaut or Goblin Sharpshooter (assuming we make them indestructible)
Infinite tap triggers for Fleshmad Steed
We can use Bitter Ordeal to have infinity sorcery spells on the stack, and Infinity library shuffles (for Cosi's Trickster I guess?
Infinity experience counters from Meren of Clan Nel Toth (which leads to infinity P/T creatures for Daxos and Kalemne, instants and sorceries costing infinity less and ezuri Putting infinity +1/+1 counters on.)
We can also get infinity spells if we have infinity creatures and Ink-Treader Nephilim or Radiate.
That's why you should have Laboratory Maniac out there too.