It is a cup for low level Pokémon between level 15 to level 20. The sum of the levels of the three Pokémon chosen cannot be more than 50. Technically, all Pokémon except for Mewtwo* and Mew are eligible to enter. However, getting strong Pokémon at that low of a level can get quite complicated. While some are catchable at low levels in certain versions (e.g. level 15 Dragonair in Pokémon Yellow), others are more complex and involve trading with in-game Trainers, then trading these Pokémon to different in-game Trainers in different games. Coincidentally, many of these involve Pokémon Yellow.
Ignoring glitches and tradebacks from Generation II, the following Pokémon may be legitimately obtained and entered into the cup.
A Pokémon with a * next to the version can only be obtained by trading across different versions, while a Pokémon with ** can only be obtained in the Japanese versions with the Japanese Blue.
In the Japanese Blue, trade a Kadabra for a Graveler at the Cinnabar Island lab. An eligible Golem cannot be obtained in the international Generation I games.
Catch a low level Golduck, then trade it for Rhydon at the Cinnabar Island labY. Trade the Rhydon for a Kangaskhan at the Cinnabar Island lab in the Japanese Blue, then trade the Kangaskhan for a Muk in the Cinnabar Island labY.
Catch a low level Golduck, then trade it for a Rhydon at the Cinnabar Island labY. In the Japanese Blue, trade the Rhydon for a Kangaskhan at the Cinnabar Island lab.
Eligible but unavailable without cheats or glitches
The following Pokémon cannot be found at a low enough level in any game from Generation I or II to be used in the Pika Cup, and none of them are available as rentals.
Several of the Trainers use underleveled Pokémon. However, some of these Pokémon become eligible with tradebacks.
Mr. Fix uses a level 15 Magnemite in Round 1, which is unobtainable without glitches or tradebacks.
Hiker uses a level 15 Seadra in Round 1, which is unobtainable without glitches.
Swimmer uses a level 17 Seadra in Round 2, which is unobtainable without glitches.
Burglar uses a level 15 Tentacruel in Round 2, which is unobtainable without glitches.
Fisher uses a level 15 Machamp and a level 15 Golem in Round 2. A level 15 Machamp is obtainable through a tradeback, but a level 15 Golem is not legitimately obtainable outside of Japan.
Some of the rental Pokémon are also underleveled. All rentals are level 15, although not all species offered can be obtained at this level.
Ponyta and Seel are offered as rentals. While they are obtainable at level 15 with tradebacks, there is no other way to obtain them at any qualifying level. Ignoring tradebacks, their lowest possible levels are 28 and 22, respectively.
Gyarados is not possible to obtain at level 15 while also knowing Hydro Pump, Leer, or Bite. While these moves are learned by Gyarados at level 1 in Red, Green, and Blue, it is not possible to catch a level 15 Gyarados in those games without glitches.
As noted above, several underleveled Pokémon are offered as rentals.
However, certain other Pokémon are not offered, despite being entirely available at level 15 without trading between games.