So if bitcoin becomes a primarily store-of-value coin, or if an altcoin that pops up in the future decides to make itself primarily for the use of storing value, will the amount of transactions be able to support miners, when the block reward drops off, and only transaction fees are left? If not enough transactions happen, what will happen to the network security? Will the big players spend their own money to secure the network to make sure the value they have in the coin is safe? Or will transaction fees be very high? Or will the network be vulnerable?...
...Also, if an altcoin is too inflationary, obviously it won't work well as a store of value. But perhaps a small amount of inflation could work; if we could estimate the amount of coins lost (through mistakes, death, or whatever else) per year, and have the emissions match that amount, we are essentially keeping the amount of coins constant (or close to it). Miners will still get rewards, and we won't end up with a grand total of 1 satoshi that isn't lost, from 21mil coins in the future.
I'm surprised to see that Peercoin hasn't already been mentioned yet. Peercoin was designed to be a store of value coin from the very beginning by Sunny King. His suggestion was that PPC should be used as a savings account or a "backbone currency". It's security model is proof-of-stake although the distribution uses an initial proof-of-work stage.
One of the measures he implemented to make sure that PPC is fit for this role was fixed transaction fees which don't go to miners/stakers but are simply destroyed and lost forever. In order to encourage staking, the coin incorporates a fixed 1% per year inflation rate instead. Hence Peercoin completely dissolves the link between transaction fees and mining/staking incentives. This means that even with very few transactions happening, PPC can still have a healthy network.
Note that this also means that PPC isn't really the type of coin that you'd use to buy your baked beans at the supermarket with.