BTC and ETH will remain #1 and #2 respectively. #3 will be heavily contested. Let's look at our candidates within the current top 10:
XRP -- Should have never come as far as #3 in the first place due to its centralized issuance, so I doubt it will be able to hold its spot.
BCH -- Will remain strong, but is basically BTC with much less support, so won't reach #3 for its lack of differentiation.
NEM -- I'll be honest, I still have no idea what they are about.
LTC -- Now that the Segwit hype-cycle has come to an end, LTC will probably not have enough differentiation to claw its way up to #3.
IOTA -- Interesting concept, but I don't see IoT taking off any time soon. If anything it seems like the hype has subsided.
NEO -- Their aim to support multiple programming languages for smart contracts looks way too ambitious to be honest.
DASH -- Looked promising, but I'm not sure whether they can keep up.
ETC -- What ETH was meant to be, but will probably starve due to lack of support.
So what's left? The Segwit2x hardfork. Should a hardfork occur, 2x will likely have more support than BCH and thus could stick around at #3. We'll have an Ethereum sandwich with Bitcoin as buns.
Interesting take. In that case, I'm curious... do you think that the Segwit2x fork will cause a genuine split? If Bitcoin users supported one of the networks (BTC, BCC or Segwit2x) over the others, wouldn't this damage the network value of each coin? It seems like these forks have created value out of thin air, but in time, I think price may more accurately reflect economic support and activity on each chain.