Universe expansion from initial explosion If the universe is expanding at an accelerated rate, is it because it hasn't reached its maximum velocity from initially exploding?
I'm thinking in terms of a bullet being fired from a weapon.  It starts at rest so there has to be a period of time for it to reach its maximum speed before beginning to slow down.
 A: The short answer is we don't know.
If you want to know what the universe will do, begin at the Einstein Field equations: $G_{\mu\nu}=8\pi GT_{\mu\nu}$. The matter that fills the universe (stars, dust, light, and any other weird stuff that hasn't been discovered yet) is contained in $T_{\mu\nu}$. The term $G_{\mu\nu}$ contains the information about the curvature of spacetime itself, which includes how it will behave (and the rate at which it will expand). 
If you make the (very reasonable) approximation that the universe is homogenous and isotropic (e.g. the same at every location and the same in every direction), you find that the universe should increase in size at a certain rate. Specifically, if you assume the universe is only filled with cold dust, you find that the size of the universe increases as $a\propto t^{2/3}$, while if you assume that the universe is filled only with radiation you find that the size of the universe increases as $a\propto \sqrt{t}$ (where a is the relative size of the universe). It is true that the universe contains both radiation and cold dust. Accounting for both species is analytically difficult, but can be solved numerically. Also, in the present universe, there is very little radiation compared to matter.
What we observe is that the universe appares to be expanding exponentially: $a\propto e^{Ht}$. This result suggests that either (1): gravity is wrong (which means that the curvature term $G_{\mu\nu}$ needs to be modified by a better model of gravity than general relativity) or (2): the manner in which spacetime curves is correct but there exists some species of undiscovered matter with really unusual properties (which would in effect be a weird form of matter in $T_{\mu\nu}$).
We know that the universe is currently undergoing an accelerated expansion, but we don't know what is the cause of this (so we dub the source of this accelerated expansion 'dark energy'). Because dark energy is unknown, we can't really make any statement on how it will behave in the future (but you can always look at a specific model of dark energy and see what it will do in the future). 
If you assume that dark energy is a new species of matter (e.g. a new term to include in $T_{\mu\nu}$), it seems reasonable that since dark energy should continue to speed up the expansion rate of the universe since dark energy was the dominant stuff that fills the universe for about 2/3 of the universe's lifetime so far, but I also have no idea on what dark energy is and how it behaves, so I can't really say whether or not it will continue behaving in the same way for a long time into the future.
Also I'd like to comment when you state "initially exploding". If we go back in time to close to the big bang, we think we understand gravity (which is general relativity) pretty well. However, we can't go all the way back. A better way to state the idea of "the big bang" is to think of the big bang as a limit where general relativity must be replaced with a theory of quantum gravity. The universe was likely very hot and dense at that time, but again, since I don't know quantum gravity I can't make to many statements about the conditions around the big bang.
A: When a bullet is fired a force acts on it, which means it gets accelerated for a certain period of time.  This happens until it reaches its maximum velocity from said acceleration, as you noted.
vf = vi + at
This equation means there is a certain period of time that the bullet will keep accelerating until reaching vf, at which point it stops accelerating (since vf is the "final" velocity).
Thus, if you believe the universe is still accelerating, then this means it hasn't reached its final velocity from the force that caused it to start accelerating.  According to Newtonian Mechanics, your conclusion is correct.
However, the question is whether or not Newtonian Mechanics can be applied to the entire universe.  Relativity and quantum mechanics have shown us that very big / very small objects are not bound by the laws of classical physics.
BTW, you mentioned how a bullet will slow down after reaching maximum speed.  This is true, but only because forces like air resistance are acting on the bullet.  If the universe reaches maximum velocity, there's no known reason it would slow down (unless you can prove there are "external" forces acting on the universe itself).  It's logical to assume it would stop accelerating, but it wouldn't necessarily lose speed.  It would probably just stay at its final velocity.
