Spoiler warning! This section of the article contains spoilers, or hints about the game's storyline or progression.
You might want to skip down to the next heading if you do not want facts about the game's storyline or plot revealed to you.
Spekkio is the master of war who appears at the End of Time. He teaches each character magic, although in a new game + you will already know it, and you can challenge him for prizes, which include rare tabs. His form and offered prizes depends on your level. You can only receive each prize set once per game.
In Nu form he's immune to all physical attacks and affected by all magic, although a double tech with a physical and magical component will hurt him.
Make sure you take each character to meet him, so he can teach them magic or describe their abilities. Robo won't be able to learn magic since he has no soul, although he can inflict Shadow type damage. Ayla won't be able to learn magic since she predates its arrival on earth, and she'll be just about useless in fighting Spekkio with her single techs anyway. Crono will learn Lightning magic from him, Lucca will learn Fire, Marle will learn Ice, and Frog will learn Water. Magus will already know Shadow magic when he joins the party.
Spekkio's form depends upon the level of whatever character is in the lead of your party. For most of the game this will be Crono, although later you can set this to whomever you want.
Frog (Levels 1-9)
Not many players are likely to ever see this form, but if you do he will give you a magic tab for beating him. He'll have 350 HP. If you want to see this form the easiest thing to do is start a new game and skip as many fights as possible, or let Crono die so he won't gain experience. It is important to note that you can beat every beginning enemy and boss without farming, and many fights will seem to be unskippable, but all you have to do is scurry along the very edges of the map to avoid them. To get past the very last part (The Factory Ruins), you will need to kill Crono and Marle/Lucca every fight with only Robo remaining so that they don't gain any exp. Robo can't learn magic so it's only Crono, Marle, and Lucca that you have to worry about.
Kilwala (Levels 10-19)
Spekkio's second form. He has 800 HP. You will win a Magic Tab and 5 Ethers for defeating him.
Have Marle heal using Aura while Crono and Lucca cast their spells on Spekkio.
Goblin (Levels 20-29)
Spekkio's third form. He has 2200 HP and will give a Magic Tab and 5 Mid Ethers for defeating him. Healing becomes important at this point and fire-resistant or absorbing equipment will help greatly. If you grind too much, your equipment won't be great enough to defeat him, but if you've already gotten past Unnatural Selection? in the story, you should be in good enough shape to beat him.
Omnicrone (Levels 30-39)
Spekkio's fourth form. He has 4800 HP and will give a Magic Tab and five Full Ethers for defeating him. You will probably want some of the most powerful single techs and some good healing magic before you attempt this form. Equipment resistant to specific elements helps.
Hit him with an onslaught of your most powerful individual or dual techs. If you have Marle in the party, cast haste on your party members. Assign a group healer and go to town.
(Recommended level 38-39)
Masamune (Levels 40-98)
Spekkio's fifth form. He has 10000 HP and will give 1 Magic Tab, 1 Power Tab, and 1 Speed Tab as well as 10 Elixirs. Finish all the sidequests to get the best equipment, and fight him whenever you are ready.
Nu (Level **/99)
Spekkio's final form. He has 20000 HP, high magic defense, and strong attacks. Defeating him earns you 10 Magic Tabs, 10 Power Tabs, and 10 Speed Tabs, as well as 10 MegaElixirs. Equipping Haste Helm and Green Dreams on everyone significantly helps with the battle, and having Lucca and Frog cast Frog Flare does a ton of damage if Frog is at low HP, which will happen a lot (Spekkio uses Halation a lot in this form). If you are playing the Nintendo DS version, Lucca's Elemental Aegis armor completely protects her from Spekkio's attacks.