Fire Red No Evos: 59 catches.

1.)Bulbasaur, Charmander, or Squirtle: Pick your starter. Pretty straightforward.
2.)Pidgey: Common encounter in Route 1/22(betwen Viridian City and Victory Road)/2.
3.)Rattata: Common encounter in Route 1/2/many others I don't care to list.
4.)Mankey: Not so common encounter in Route 22, but it's not very rare.
5.)Spearow: Encountered in many routes as early as Route 3(near Mt. Moon).
6.)Caterpie: Common encounter in Viridian Forest.
7.)Weedle: Same as Caterpie. Both have the same encounter rate in both Fire Red and Leaf Green.
8.)Nidoran Male: Slightly rare encounter in Route 3 for Fire Red players. As rare as 1% in Leaf Green.
9.)Magikarp: Can be gotten as early as buying it from the merchant in the Mt. Moon Pokemon Center. If you don't want to spend money, just wait until you get the Old Rod in Vermillion City and I can guarantee you'll fish lots of Magikarp.
10.)Zubat: Common encounter in Mt. Moon and many caves.
11.)Geodude: Another common encounter in Mt. Moon and other caves.
12.)Paras: Encounter-able in Mt. Moon at a 100% rate in B1F. Rare in other floors.
13.)Ekans: Somewhat uncommon encounter in Route 4. More common in Route 11. Fire Red exclusive. Leaf Green players get Sandshrew instead(same probabilities and encounter places apply).
14.)Oddish: Somewhat uncommon encounter in Route 24. More common in the routes near Silence Bridge but I needed the guy as early as Cerulean. Fire Red exclusive and Leaf Green players instead get Bellsprout.
15.)Abra: Rare encounter in Route 24. Good luck. You could also buy one at the Game Corner as one of the cheaper mons
16.)Meowth: Common encounter in Routes 5 and 6.
17.)Nidoran Female: Trade your Nidoran Male with a lady in the Underground Path for Fire Red players. Leaf Green players can instead trade a Nidoran Female for a Nidoran Male. Works out in the end.
18.)Diglett: Literally the most common encounter in Diglett's Cave. You're likely to run into one if you go for Flash.
19.)Drowzee: Route 11 encounter. 25% isn't so rare.
20.)Voltorb: Most common Route 10 encounter.
21.)Machop: Rock Tunnel encounter.
22.)Onix: Rock Tunnel encounter.
23.)Growlithe: Rarest encounter in Route 7 and 8. More common in 8. Fire Red exclusive. Leaf Green players instead get Vulpix. Sadly this means you sacrifice an Intimidate mon for a slightly bulkier Fire type. Tauros gets Intimidate if you allow it.
24.)Eevee: Gift mon at the Celadon Mansion roof. Take the back entrance.
25.)Doduo: Common encounter in Route 16.
26.)Mr. Mime: Trade Abra with a dude somewhere on the right half of Route 2. Requires either a trip through Diglett's Cave or I guess you could fly to Viridian and just Cut a tree.
27.)Gastly: Really common encounter in Pokemon Tower.
28.)Cubone: Rare encounter in Pokemon Tower. More common on the 7th floor.
29.)Snorlax: There are two Snorlax you can catch. One is on Route 12, the other is on Route 16. You need to get past one to get to Fuchsia. Catch whichever you prefer. I personally prefer the one on Route 12 since the Silence Bridge gives good experience and is not the worst rematch spot.
30.)Horsea: Common Super Rod encounter in a bunch of routes, particularly the Silence Bridge ones(12 and 13). Super Rod is gotten on Route 12. A bunch of weird crap but basically that 84% encounter is only for Fire Red and Krabby is the one Leaf Green players get instead. Neither is a version exclusive though.
31.)Psyduck: 1% of the time appears on Route 12 with a Super Rod. Or you could just surf in the Fuchsia pond for a 100% encounter rate. Fire Red exclusive. Leaf Green players instead get Slowpoke.
32.)Venonat: Not so rare encounter on Routes 12, 13, 14, and 15.
33.)Ditto: Most commonly found in Route 15(15% encounter rate), but also appears in Pokemon Mansion at 10% encounter rate.
34.)Rhyhorn: Technically the most common Safari Zone exclusive encounter. Also not the hardest to catch. Nidorans are more common but they are far from Safari Zone exclusive.
35.)Tangela: Grass under Pallet Town always spawns these.
36.)Tentacool: Literally 100% encounter rate when you surf on the routes that go to Cinnabar.
37.)Omanyte or Kabuto: Pick the correspondent fossil at Mt. Moon and revive in Cinnabar Island's lab. Your choice.
38.)Aerodactyl: Get the Old Amber in the Pewter Museum's back entrance. Requires Cut to get. Revive in Cinnabar Island's lab.
39.)Koffing: Common encounter in the Pokemon Mansion. More common in Fire Red where Grimers are rare. Leaf Green players instead find Grimer more commonly but Koffing more rarely.
40.)Goldeen: Common Good Rod encounter in the Fuchsia pond. Probably also many other places.
41.)Poliwag: Less common Good Rod encounter in the Fuchsia pond.
42.)Krabby: Spawn about 20% of the time with the Good Rod in Route 19(south of Fuchsia). Whereas Horsea is the common one in Fire Red, Krabby is the common one in Leaf Green. Neither is exclusive.
43.)Shellder: Common Super Rod spawn on the water south of Pallet Town. Fire Red exclusive. Leaf Green players instead get Staryu.
44.)Magnemite: Most common Power Plant encounter. Be careful to not fight Magnemite with Magnemite.
45.)Seel: Seafoam Islands. No other way about it. As rare as 10% in B1F but as common as 40% in B4F and 50% in B5F
46.)Dratini: I just prefer to buy it from the Game Corner but it's also a Safari Zone encounter in the water. 15% but it's a fished Pokemon so you don't waste steps. Whichever pleases you.
47.)Scyther: Rare encounter in Safari Zone(as with the many exclusives). This one you can buy in the Game Corner. If you want to try the Safari Zone, it's most commonly found in the Center Area(read the signs)(it's the entrance area). Leaf Green players instead get Pinsir.
48.)Porygon: Buy it in the Game Corner. No way around that. Incredibly pricy in both versions, but Leaf Green players are lucky and only have to pay 6,000 coins instead of the coin cap(9,999) that Fire Red players have to.
49.)Exeggcute: Technically as common as Rhyhorn in the Safari Zone. Lower catch rate though. Rare Berry Forest encounter. Safari Zone encounter is tempting but Berry Forest one can be put to slept and have its health lowered. Do whichever one pleases you.
50.)Kangaskhan: Another rare Safari Zone encounter. Most commonly found in Area 1(read the signs)(east of Center Area).
51.)Tauros: Safari Zone. Most common in Area 3(read the signs)(it's the one with the Surf house).
52.)Chansey: Safari Zone. Most common in Area 2(read the signs)(north of Center Area, northwest of Area 1).
53.)Lapras: Random Silph Co. guy gives it to you. He's found in the room where you fight your rival.
54.)Grimer: Rare Pokemon Mansion encounter. More common in Leaf Green, but there Koffing is the rare one. 5%, good luck.
55.)Ponyta: Apparently the most common Kindle Road encounter. My game thought otherwise, but I probably just have bad luck.
56.)Farfetch'd: Trade Spearow for one on Vermillion. I'd personally keep the Spearow but you can always catch more Spearows and Doduo also exists. And I do need the catches, so eh.
57.)Articuno: Bottom of Seafoam Islands(http://www.ign.com/wikis/pokemon-firered-leafgreen-version/Seafoam_Islands)(http://www.psypokes.com/frlg/gamemaps.php?view=seafoam). Low catch rate.
58.)Zapdos: At the end of the Power Plant(http://strategywiki.org/wiki/Pok%C3%A9mon_FireRed_and_LeafGreen/Power_Plant). Low catch rate.
59.)Moltres: Top of Mt. Ember(http://www.ign.com/wikis/pokemon-firered-leafgreen-version/Mt._Ember). Low catch rate.
60.)Sorry, there are only 59 legal catches, and I had to stretch the rules to allow catching legends/non-evolvers but not use them on my team. Evolve something(that you won't use on your team or else that definitely goes against the challenge rules.) or catch some evolved Pokemon. Me? I flubbed and got a Pikachu catch accidentally. So that's how that is.

Edit: Fixed the mistake where I said Venonat was a FireRed exclusive and Slowpoke was the LeafGreen counterpart. It's actually Psyduck.