Edit: Err, I was supposed to get mod permission? Sorry.
(Really old subject)
http://www.smogon.com/ingame/misc/adv_frlg_ingametiers
I just ran through a game of Leaf Green. I have to say, just as I thought, Mr. Mime was a godsend.
No, listen!
For someone without access to trading with Pokemon FireRed or LeafGreen, there is no trading before the Elite 4. Which automatically reduces its only true competition in pure-Psychics to Kadabra.
Here's where Kadabra has issues, and some of these apply to Alakazam as well.
1.) Kadabra has a REALLY shallow movepool. No elemental punches, Hidden Power is unreliable. It gets Psychic, Recover, and Trick. YES, recover is nice, but it is not everything. AND Calm Mind isn't even available without a TM on Kadabra.
Mr. Mime, however, gets the re-buyable Thunderbolt, which is nice for beating up Gyarados and other waters in the Elite 4 if you'd like. Mr. Mime also gets (yes, laugh at this one) Magical Leaf, in time to hit Rock Tunnel. If you start with Charmander, and don't have a grass or water type (you probably won't), this will help against the plethora of rock / ground types.
2.) Mr. Mime has better staying power with better Defense and Special Defense. Kadabra is faster and stronger, but Mr. Mime is usually able to OHKO anything Kadabra can.
3.) Abra is gotten after Nugget Bridge, which, if you're like me, is after Cerulean gym. Mr. Mime, on the other hand, is only about 10-20 minutes later, on the other side of two tunnels, comes with an actual move to fight with, AND levels up 50% faster for being in a trade.
4.) Trick is gotten at a much higher level with Kadabra, which is not terrible per se, but a few levels difference is a question of "do I get that thick club sonner or later?" or "do I really have to waste my money catching Abra for TwistedSpoons".
My argument is that Mr. Mime should definitely go above Kadabra and Alakazam pre-Elite 4. I also argue that it should go above Jynx. With Jynx, you not only get it MUCH later (as in, after Lavender Town and the Ghosts, and after Celadon with its plethora of poison-types that you still don't have a real advantage over), you also have to waste the Psychic TM (non-renewable - Jynx does not even get a single Psychic STAB move without a TM), and have to STILL BUY the Ice Beam TM (rather than a natural level-up). And a sleep move, while nice, is not always necessary in-battle.
Starmie has a similar problem to Jynx, in not getting Psychic, but it makes up for it in recovery moves and other options.
Post-Elite 4, you have more freedom - Kadabra can get the elemental punches via breeding, while Mr. Mime still only gets Thunderbolt (I was surprised too, when I saw he couldn't get Ice Punch via breeding, and we'll ignore traded tutors).
For the reasons summarized above, I believe Mr.Mime should be moved to Top Tier, and reconsider Abra's placement, possibly even that of Jynx for in-game.
Mr. Mime
Top Tier
Availability: Early
Stats: Above Average
Movepool: Above Average where it counts.
Major battle match-ups: Rock Tunnel is good with Confusion and Magical Leaf; pokemon Tower is excellent against ghost-types. Sweeps Erika, Koga handily. Survives Sabrina and Blaine, and can hit back with neutral attacks. Sweeps Giovanni fairly quickly, as well as the Team Rocket Saga. Thunderbolt hits hard against Lorelei's team, and some of Lance's, Psychic hits Bruno, Agatha fairly hard. Can help with rival's team, though nothing spectacular.
Other Comments: Gotten fairly early, before the SS Anne, but does require a detour. Gets Psychic through level-up, and can add diversity with Thunderbolt and Magical Leaf. Gets Trick for utility and stealing items like Thick-Club and mushrooms. Nominally weaker than Kadabra; no sleep move, no secondary type, but it packs a punch where it counts. Levels up quicker from trade which is a double-edge sword if it doesn't obey.
Lastly:
Dratini - Bottom Tier is really low. You can get him in Celadon, which is earlier than most pokemon. He's expensive, yes, but level him up a bit, and his stats are GREAT, and he gets LOADS of TMs. As a Dratini / Dragonair, he Erika, Blaine, and with Surf, it can take down Giovanni. I'd say at least Low-tier, maybe Mid. And if Tauros is low-tier...come on, you know this thing is better than that. Okay, no STAB normal moves, but you know Draggy's good for it!
Dratini
Mid Tier
Availability: Mid-game
Stats: Above Average
Movepool: Above Average
Other Comments: Can be bought fairly early. If you have enough money, and get the elemental TMs immediately, he'll shine. Downside is that he evolves fairly late, but Dragon has some nice resistances by itself. Can compensate for whatever your team is lacking, VERY versatile
Major battle match-ups: (If gotten early enough): Erika is swept with flamethrower or ice beam, if you purchased it. Koga will not enjoy special moves much. Sabrina is an unremarkable matchup. Decimates your rival with elemental moves from the start. Resists Blaine's fire moves, and can sweep with Surf. Giovanni is beaten handily with Surf. Against Lorelei, you've gotta be faster or your dead. If you've evolved it to Dragonite, you'll resist Bruno's moves except Rock Tomb, which can stack up. Lance's Gyarados will go down quickly, but Aerodactyl (Ancient Power) and Dragonair / Dragonite (Outrage) are all super-effective. Ice beam, you're fast enough, will hurt them worse than they hurt you.
Credits go to Bulbagarden for images, thank you. And Smogon, et al. for the original tier list, and hosting the forum.
(Really old subject)
http://www.smogon.com/ingame/misc/adv_frlg_ingametiers
I just ran through a game of Leaf Green. I have to say, just as I thought, Mr. Mime was a godsend.
No, listen!
For someone without access to trading with Pokemon FireRed or LeafGreen, there is no trading before the Elite 4. Which automatically reduces its only true competition in pure-Psychics to Kadabra.
Here's where Kadabra has issues, and some of these apply to Alakazam as well.
1.) Kadabra has a REALLY shallow movepool. No elemental punches, Hidden Power is unreliable. It gets Psychic, Recover, and Trick. YES, recover is nice, but it is not everything. AND Calm Mind isn't even available without a TM on Kadabra.
Mr. Mime, however, gets the re-buyable Thunderbolt, which is nice for beating up Gyarados and other waters in the Elite 4 if you'd like. Mr. Mime also gets (yes, laugh at this one) Magical Leaf, in time to hit Rock Tunnel. If you start with Charmander, and don't have a grass or water type (you probably won't), this will help against the plethora of rock / ground types.
2.) Mr. Mime has better staying power with better Defense and Special Defense. Kadabra is faster and stronger, but Mr. Mime is usually able to OHKO anything Kadabra can.
3.) Abra is gotten after Nugget Bridge, which, if you're like me, is after Cerulean gym. Mr. Mime, on the other hand, is only about 10-20 minutes later, on the other side of two tunnels, comes with an actual move to fight with, AND levels up 50% faster for being in a trade.
4.) Trick is gotten at a much higher level with Kadabra, which is not terrible per se, but a few levels difference is a question of "do I get that thick club sonner or later?" or "do I really have to waste my money catching Abra for TwistedSpoons".
My argument is that Mr. Mime should definitely go above Kadabra and Alakazam pre-Elite 4. I also argue that it should go above Jynx. With Jynx, you not only get it MUCH later (as in, after Lavender Town and the Ghosts, and after Celadon with its plethora of poison-types that you still don't have a real advantage over), you also have to waste the Psychic TM (non-renewable - Jynx does not even get a single Psychic STAB move without a TM), and have to STILL BUY the Ice Beam TM (rather than a natural level-up). And a sleep move, while nice, is not always necessary in-battle.
Starmie has a similar problem to Jynx, in not getting Psychic, but it makes up for it in recovery moves and other options.
Post-Elite 4, you have more freedom - Kadabra can get the elemental punches via breeding, while Mr. Mime still only gets Thunderbolt (I was surprised too, when I saw he couldn't get Ice Punch via breeding, and we'll ignore traded tutors).
For the reasons summarized above, I believe Mr.Mime should be moved to Top Tier, and reconsider Abra's placement, possibly even that of Jynx for in-game.

Mr. Mime
Top Tier
Availability: Early
Stats: Above Average
Movepool: Above Average where it counts.
Major battle match-ups: Rock Tunnel is good with Confusion and Magical Leaf; pokemon Tower is excellent against ghost-types. Sweeps Erika, Koga handily. Survives Sabrina and Blaine, and can hit back with neutral attacks. Sweeps Giovanni fairly quickly, as well as the Team Rocket Saga. Thunderbolt hits hard against Lorelei's team, and some of Lance's, Psychic hits Bruno, Agatha fairly hard. Can help with rival's team, though nothing spectacular.
Other Comments: Gotten fairly early, before the SS Anne, but does require a detour. Gets Psychic through level-up, and can add diversity with Thunderbolt and Magical Leaf. Gets Trick for utility and stealing items like Thick-Club and mushrooms. Nominally weaker than Kadabra; no sleep move, no secondary type, but it packs a punch where it counts. Levels up quicker from trade which is a double-edge sword if it doesn't obey.
Lastly:
Dratini - Bottom Tier is really low. You can get him in Celadon, which is earlier than most pokemon. He's expensive, yes, but level him up a bit, and his stats are GREAT, and he gets LOADS of TMs. As a Dratini / Dragonair, he Erika, Blaine, and with Surf, it can take down Giovanni. I'd say at least Low-tier, maybe Mid. And if Tauros is low-tier...come on, you know this thing is better than that. Okay, no STAB normal moves, but you know Draggy's good for it!

Dratini
Mid Tier
Availability: Mid-game
Stats: Above Average
Movepool: Above Average
Other Comments: Can be bought fairly early. If you have enough money, and get the elemental TMs immediately, he'll shine. Downside is that he evolves fairly late, but Dragon has some nice resistances by itself. Can compensate for whatever your team is lacking, VERY versatile
Major battle match-ups: (If gotten early enough): Erika is swept with flamethrower or ice beam, if you purchased it. Koga will not enjoy special moves much. Sabrina is an unremarkable matchup. Decimates your rival with elemental moves from the start. Resists Blaine's fire moves, and can sweep with Surf. Giovanni is beaten handily with Surf. Against Lorelei, you've gotta be faster or your dead. If you've evolved it to Dragonite, you'll resist Bruno's moves except Rock Tomb, which can stack up. Lance's Gyarados will go down quickly, but Aerodactyl (Ancient Power) and Dragonair / Dragonite (Outrage) are all super-effective. Ice beam, you're fast enough, will hurt them worse than they hurt you.
Credits go to Bulbagarden for images, thank you. And Smogon, et al. for the original tier list, and hosting the forum.