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'll already know it, and you can challenge him for prizes, which include rare tabs. His form (and thus the prizes he offers) depends on your level. Sadly you can only receive each prize set once per game. He stops gaining new forms at around level 40, but if you get up to level **/99, he'll turn into an ultimate Nu form.
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 shows up obviously.
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'll 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.
Kilwala (Levels 10-19)
The next level up of Spekkio, you'll win a magic tab and 5 ethers. He has 800 HP, so 7-8 magic casts will defeat him.
Have Marle heal using Aura while Crono and Lucca cast their spells on Spekkio.
Goblin (Levels 20-29)
2200 HP this time, he'll give a magic tab and 5 mid ethers. This is where he really starts to get hard, you'll need to be actively healing to have a chance, and fire-resistant or absorbing equipment wouldn't hurt either. 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 walkthrough, you should be in good enough shape to beat him.
Omnicrone (Levels 30-39)
Spekkio has 4800 HP this time and now taking after the thugs from Guardia castle. You'll win a Magic Tab and five Full Ethers for beating him. You'll 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 too.
Masamune (Levels 40-98)
Once you've gone through your first game and are in a new game plus, this is how you'll see Spekkio most of the time. He gets easier to beat as you level up, but stay tuned, there's more to come. He has 10000 HP, gives a Magic, Power, and Speed tab as well as 10 Elixirs. Just finish off all the sidequests to get the best equipment, and fight him whenever you're ready.
Nu (Level **/99)
If you max out your level, you'll finally get to see Spekkio's ultimate form, which also happens to be the 2nd hardest fight in the game (Next to Dream Devourer, which only appears in the Nintendo DS release of the game). He has 20000 HP, high magic defense, and a lot of pulverizing attacks even though you'll be maxed out on everything. If you win you get 10 Magic, Power, and Speed Tabs, as well as 10 MegaElixirs, and enough of his praise to last a year. Haste Helm can help the fighting go faster, equipping Green Dreams on everyone improves your odds, 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 Hallation a lot in this form). If playing the Nintendo DS version, Lucca's Elemental Aegis armor renders this fight unloseable.