A router does nothing but route. But as a guess take the number of routes on the internet and that would be a good guess.
Looks like the Internet routing table is at 178,000 networks/prefixes. These are actively announced routes/paths on the internet. Basically a router saying "hey, wanna get here? Here's how"
If I were to take an educated guess at "main path" routers....probably a couple thousand.
----output from a well known route server, cookie for anybody that knows who owns it....c'mon, it should only take you a few seconds *cough* 12. *cough*---------
oh, and end the drama please...we're all friends here. Plus it's been awhile since I've been deep into BGP so talking about it helps jog the mind.
route-server>sh ip bgp sum
BGP router identifier 10.1.2.5, local AS number 65000
BGP table version is 9540424, main routing table version 9540424
178820 network entries using 18060820 bytes of memory
3397354 path entries using 163072992 bytes of memory
41447 BGP path attribute entries using 2321816 bytes of memory
37440 BGP AS-PATH entries using 971332 bytes of memory
4 BGP community entries using 96 bytes of memory
0 BGP route-map cache entries using 0 bytes of memory
0 BGP filter-list cache entries using 0 bytes of memory
BGP using 184427056 total bytes of memory
Dampening enabled. 2147 history paths, 499 dampened paths
178614 received paths for inbound soft reconfiguration
BGP activity 288940/110120 prefixes, 9392009/5994655 paths, scan interval 60 secs
Neighbor V AS MsgRcvd MsgSent TblVer InQ OutQ Up/Down State/PfxRcd
10.0.0.2 4 7018 0 0 0 0 0 never Idle (Admin)
12.0.1.63 4 7018 2932482 71199 9540404 0 0 7w0d 0
12.123.1.236 4 7018 1976464 71198 9540418 0 0 7w0d 178699
12.123.5.240 4 7018 2035050 71199 9540418 0 0 7w0d 178695
12.123.9.241 4 7018 2006059 71201 9540418 0 0 3w6d 178699
12.123.13.241 4 7018 2452605 71202 9540418 0 0 3w6d 178702
12.123.17.244 4 7018 2046156 71199 9540418 0 0 7w0d 178696
12.123.21.243 4 7018 2270460 71199 9540418 0 0 7w0d 178697
12.123.25.245 4 7018 2590910 71199 9540418 0 0 7w0d 178700
12.123.29.249 4 7018 2107890 71199 9540418 0 0 7w0d 178696
12.123.33.249 4 7018 2667659 71198 9540418 0 0 7w0d 178694
12.123.37.250 4 7018 2882204 71201 9540418 0 0 3w6d 178699
12.123.41.250 4 7018 2306119 71199 9540418 0 0 7w0d 178700
12.123.45.252 4 7018 2580638 71202 9540418 0 0 3w6d 178702
12.123.133.124 4 7018 2045452 71198 9540418 0 0 7w0d 178702
12.123.134.124 4 7018 2164216 71201 9540418 0 0 3w6d 178701
12.123.137.124 4 7018 2447154 71201 9540418 0 0 3w6d 178695
12.123.139.124 4 7018 2358867 71198 9540418 0 0 7w0d 178701
12.123.142.124 4 7018 2614348 71198 9540418 0 0 3w6d 178702
12.123.145.124 4 7018 2692404 71199 9540418 0 0 3w6d 178699