Is the TTL I see on a packet arriving from a server to my computer is the original TTL or the TTL after the routers between stripped some numbers?

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Each IP node sets an initial TTL when sending a packet. Typical initial TTLs are 255, 128 and 64. A few IP implementations use other values.

When a router forwards a packet the TTL value is reduced by 1. The packet will be dropped, when TTL reaches zero.

If you receive a packet with a TTL of 240, it was likely forwarded by 15 routers. If you receive a packet with a TTL of 120, it was likely forwarded by 8 routers or 125 routers. Then again, I have never encountered a network with a diameter of more than 30 hosts.

Please note, that the TTL will not be reduced while the packet is forwarded through IPsec and similar tunnels.