First chip with 1000 or more transistors?
Based on the datamath webpage there were ICs around with more than 1000 transistors as early as 1967 used in TIs calculators prototypes.

But it's not clear if this is a single chip or all three had together 1000 transistors and the exact chip is not mentioned.

I've done some further research on the CalTech prototype. There is a patent available: Patent US3819921 which describes the calculator prototype from TI.

Figure 15, 16, 17 and 18 show the logic of the four implemented ICs. The biggest one in figure 15 has 151 NAND gates. Figure 20 shows the construction of a single NAND gate. As can be seen, they are made using 6 PNP and 2 NPN transistors, so each NAND gate needs 8 transistors.

For the chip with 151 gates this results in a total transistor count of 1208.

Now that patent was filed in 1971, which is later than the original prototype, so I'm not 100% sure that the early prototype used already this chip. The patent mentions an application number 671,777 filed in Sept. 29, 1967. I've found a document where the original patent application and it's claims get discussed but as the whole description is not available, it's hard to say if the ICs were altered during the 4 years. Claims 15 and 35 speak of a "large plurality" of same gates, so that might be an indication that the ICs were already the same.

Based on the computer history webpage in 1965 there was a MOS ROM with 1024 bits developed from General Microelectronic. As a MOS ROM it would have needed 1 transistor per bit (if I'm not mistaken), so that chip should have had 1024, sounds like maybe the first as only 2 companies were working on MOS at that time (based on this). I haven't found a supporting source for that claim though.


  In mid- 1965 only two companies were producing MOS  ICS — General
  Microelectronics and General Instru-  ment—but interest in the new
  process was high.


Also mentioned in the Electronicsmagazine Special Commemerative Issue on page 251 is a MOS device for the military containing 968 components. Probably not all of them are transistors, but it seems like it was doable in 1965. Reading further in that document they claim that a 1024 bit ROM was announced by Philco-Ford in 1968 (page 271):


  A 64-bit MOS memory organized into 16 4-bit words was offered by
  Fairchild. A year later Philo-Ford announced a 1,024-bit ROM.