Scavenger Hunt

[AlltheTokens]

Can anyone find me a card that would get higher than +5/+5 from Embiggen.

I have found a few that have 5 types.  Inkmoth Nexus is a Land Artifact Creature Phyrexian Blinkmoth.  Calix, Guided by Fate is Legendary Enchantment Creature Human Druid.  Seton's Scout is a Creature Centaur Druid Scout Archer.

[Landale]

Burakos, Party Leader gets +7/+7. Legendary Creature Orc Cleric Rogue Warrior Wizard
If he's your commander with Raised By Giants out, he'll get +8/+8 thanks to that adding Giant onto his types.

Stonework Packbeast also gets +7/+7 as an Artifact Creature Beast Cleric Rogue Warrior Wizard, but does not have a reliable way to tack on another type.
Tajuru Paragon and Veteran Adventurer get +6/+6 being Creature (Elf/Human) Cleric Rogue Warrior Wizard.

[UrizenII]

Correct me if I'm wrong, but I believe Shapeshifters take the cake here because they are every creature type at all times (I know Changeling has weird interactions sometimes, but it should still count).  Faceless Haven would be the best because it is a snow land and has an ability that animates it, which says it becomes a creature with all creature types that's still a land.  Plus, if Changelings don't actually count here, Haven's ability doesn't give it Changeling, it just says it has all creature types.

From what I can find, as of OTJ, there are currently 299 creature types, so your Embiggen bonus would be +302/+302 on an animated Faceless Haven (1 for supertype snow, 2 for card types land and creature, and 299 for each creature type).

[Landale]

Correct me if I'm wrong,

"Non-Brushwagg". The second a creature gets that type is the second it becomes an illegal target and gets nothing from Embiggen.

[UrizenII]

"Non-Brushwagg". The second a creature gets that type is the second it becomes an illegal target and gets nothing from Embiggen.

Oops.  I knew it sounded too good to be true.  Well, I did at least find one other that gets +7/+7: Gideon, Ally of Zendikar.  With his +1 ability, he becomes a Legendary Planeswalker Gideon Creature Human Soldier Ally.

I don't think you can get bigger than +7/+7 with just a single card.  You'd need others to add additional types, like Liquimetal Torque to turn it into an artifact.  Ashaya, Soul of the Wild would turn the creature into a land as well, Enchanted Evening would add the enchantment type, and Rimefeather Owl adds the snow supertype.

[Landale]

I don't think you can get bigger than +7/+7 with just a single card. 

I don't think so either.
I had brought up Raised by Giants earlier because that is the most reliable way I can find to be able to consistently guarantee +8/+8, since it'll allow Embiggen to be in a Burakos, Party Leader deck in the first place and you'll always have access to both of them.