Topps Archives has tons of Big Red Machine cards. It’s got Big Red Machines out the wazoo, BRM’s out the yin-yang! No guys from the 1990 World Series team however. I’ll do this set by set (ending with an overview of player by player).
Base Set
Despite 5 of the 9 Big Red Machine members being included in the product, there are actually only 2 guys in the Archives base set. The first being Joe Morgan, who has a card in the 1954 subset, and the second being Ken Griffey Sr., who has a Fan Favorite SP on the 1977 design.

Normally, I’d show pictures of what cards are in the original sets, but there are so many options here that I won’t. If you factor in the SP cards from 201-240, there are so many years that I’d end up including almost every year they had cards anyways. That said, here’s a breakdown of who is in the originals from the 4 main years of the Archives set:
1954 Topps – None
1971 Topps – 8 of the 9 (missing Griffey) and 2 subset cards:
Sparky Anderson MG, Johnny Bench, Tony Perez, Joe Morgan, Dave Concepcion RC, Pete Rose, Cesar Geronimo RC, George Foster RC, NL HR Leaders (Bench/Perez), NL RBI Leaders (Bench/Perez)
1980 Topps – all 9 and 1 subset card:
Detroit Tigers CL (Sparky MG), Bench, Perez, Morgan, Concepcion, Rose, Griffey, Geronimo, Foster, Rose HL (10th season 200+ Hits)
1984 Topps – 7 of the 9 (missing Sparky and Bench) and 6 subset cards :
Perez, Morgan, Concepcion, Rose, Griffey, Geronimo, Foster, Superstars Retire HL (Bench along w/Yaz & Perry), NL Active AVG Leaders (Rose), NL Active RBI Leaders (Perez), NL Active HR Leaders (Perez), NL Active Hit Leaders (Rose, Perez), NL Active SB Leaders
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Parallels
Topps only included parallels of the first 200 cards, so Morgan is the only player with a Gold parallel or a printing plate from the base cards.

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Fan Favorite Autographs
The SP card of Griffey Sr. from the 1977 design also appears in the autographed fan favorites. Additionally, George Foster is the 3rd Big Red Machine member in this product, as he has a Fan Favorite Auto from the 1978 design.

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Reprints
The reprints bring it up to 5 BRM members, as Topps issued reprints of Sparky Anderson’s rookie card (1959 – from his playing days), the first single player card for Johnny Bench (1969). Additionally, Joe Morgan’s cards from 1967 and 1973 are included.


Since there are printing plates for each of these, that’s 4 more cards with 4 printing plates #/1.

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1969 Deckle Edge
Joe Morgan didn’t have a Deckle Edge in the original 1969 set, but Topps gave him a card in this year’s Archives set. Unfortunately, they used a photo from his 1980 stint with Houston, not the stint from the late 1960’s from the era of this set. As with all inserts in this product, there’s 4 printing plates for this card, too.

Original – Pete Rose was the only BRM-er with a card from the original Deckle Edge set.

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1977 Cloth Stickers
They definitely love Joe Morgan in this product – he and Bench both have a card in this insert set.

This again means there are 2 more sets of printing plates out there. I found a picture of one of Bench’s from eBay – the printing on this one looks really beat up.

Original – Bench, Perez, Morgan, and Rose all have cards in the original 1977 Cloth Sticker set. I’m kind of surprised Foster, who won the ’76 All-Star game MVP and was en route to smashing 52 homers and the the NL MVP at the time these were released.


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1958 Classic Combinations
Topps matched Joey Votto up with Johnny Bench here. A good choice, I must say. There actually isn’t a printing plate inserted of these cards – they didn’t do this for the retail only inserts.

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Autographed Originals
There is an exchange for a buyback autograph of a Johnny Bench original card. These were supposedly numbered out of 5.
So, all told – if you wanted to collect every single Big Red Machine card I mentioned above, you’d have 46 cards to find. If you didn’t want the 1/1’s – that number would be 14, and if you didn’t want any parallel versions of cards, you’d have 13 (I’m not counting the relics of the cloth sticker as parallels, though you could make the argument they are).
George Foster (1) – FF Autograph
Griffey Sr. (2) – FF SP, FF Autograph
Sparky Anderson (5/1) – Reprint, 4 Printing Plates
Bench (12/4) – Reprint, Cloth Sticker, Classic Combination, Autographed Original, 8 Printing Plates
Morgan (26/6) – Base, Gold Foilboard, 2 Reprints, Deckle Edge, Cloth Sticker, 20 Printing Plates