Why can't we run the laws of physics backwards and forwards in time infinitely? So assuming we know all the laws of physics in differential equation form, and I have an estimate for the current large scale state of the universe (whatever standard assumptions/data cosmologists use about the current large scale state of the universe in order to extrapolate the state of the universe on the large scale far into the future or far into the past... whatever standard assumptions are used to estimate that there was a big bang in the past)
It seems to me that I could plug these into my differential equations and find out the state of the universe infinitely far back or infinitely in the future.
So why couldn't I plug in a time 100 billion years before today (before the big bang) and find out the state of the universe far before the big bang?
Is there something in the theory/mathematics that forces the equations to begin at a certain time t(big bang)... and not allow us to extrapolate prior to that?
 A: Your accepted answer explains why we cannot do it today - mostly because we simply are not capable to solve the equations we came up with, and even for incredibly simple examples like ${dx}/{dt}=-x^2$ the resulting formulas are already nixing any chance of success. In a sense this is not what you are asking: obviously as another of your assumptions you are proposing is that we know the complete state of the universe at a time t, we are already in fantasy land (or let's say, the question would maybe be better placed in Philosopy.SE).
So this answer considers that your assumptions are, in fact true:

*

*We have (at some point in the future) all the theories, and due to our diligence or good luck they happen to be both true and complete. We know how the universe works, period. Nothing in our (future) formulas is wrong, we have the complete algorithm.

*We somehow manage to record the complete state of the universe at a point $t$ in time.

*And we find a mathematical way to work with the kinds of singularities, as mentioned elsewhere, or the brand new, complete formulas are, in some unfathomable way, immune or free from them. Heck, maybe we find a way to simulate the universe one Planck time into the future or past, and somehow this not-so-general set of algorithms does not suffer the "blowing up" problem.

The obvious answer of why this is not possible, and never will be, is that to record the complete state of the universe, we need to store it somewhere, or hold it in some form of memory. This storage will again be part of our physical universe. So where do we store the fact that we have stored the state of the universe? Do we need to add some space to our storage system for that? Where do we store the information that we did that? And so on and so forth. No system can ever store the complete state about itself in itself, out of principle.
But OK. Let's waive this issue, and assume we can store the info in some interdimensional pocket or in a separate universe.
Then let's also assume that the internal workings of the universe are, indeed, open to calculations - i.e., assume there are no completely random, nondetermined events anywhere (right down to the Planck scale). Then what would we use to calculate the state of the universe at a future time (even if it's just a Planck time "tick" in the future)? You would need a computing device which not only is able to hold the instantaneous state of the universe in memory, but also be able to do all the calculations - some of them may be able to be abstracted (i.e. the flight path of galaxies in this very short term); but we still will have to simulate every single particle right down to the quantum level because we know that quantum effects can bubble up into the visible scale. So we cannot just abstract everything away - we need a kind of computing device which is... like... an universe. So, you would need a second universe which "runs" the current universe.
Then the question is how fast this computation can run, and you can of course see where this is going - this computing device will by force run slower than the actual universe, so if you might be able to kind of "walk" the state forward, you will never ever be able to use it in any sense useful, i.e. to do proper predictions or backward calculations.
I'll handwave any mathematical resistance from the likes of Gödel's Incompleteness Theorem or other like-minded problems (which are not a sign of our current knowledge being incomplete or maybe wrong, but a real, final, show-stopper). Or chaos theory, which shows that even utterly trivial systems can very quickly show emergent chaotic behaviour that is more or less intractable in closed form.
Hey, and if we were able to overcome all these problems, a new problem occurs once we are able to roll the universe forward or backward: any information we get from this will feedback into our current time. So we get endless recursion - this feedback will need to go into the simulation, which will influence the simulation, which will be an endless, deep cycle we can by definition never get out of. The calculation would grind to a halt immediately, no matter how fast our computer is.
To sum it up: even with optimum conditions including a good amount of SciFi, it will always be practically impossible to run the universe in a simulation, which is what it means if you are talking about running it backward or forward.
And obviously, many of the assumptions I used (the existence of a closed-form set of algorithms for the universe's mechanics; the advance of maths to be able to actually handle the formulas; the non-existence of true randomness; the possibility to even "scan" the current state; the existence of parallel universes for storage and so on; the discreteness of reality, i.e. whether reality runs in "ticks" or not) are either unlikely or just so far out that it's hard to find words for it. Any of it not being true would make it not only practically, but theoretically completely impossible.
A: I'll take a bit of a philosophical angle here, and just discuss part of your opening:

So assuming we know [...], and I know the current state of the universe completely.

That is a fundamentally impossible assumption. Not for practical reasons, not because of quantum mechanics, but because you are part of the universe, and you "knowing" something must be an attribute of your state. What you are requiring is that a small part of the universe, you, somehow encodes the state of the whole universe; moreover this must happen not by accident, not by divine intervention from outside the universe, but presumably on purpose by you arranging to acquire knowledge about the whole universe, all the time remaining inside the universe. I don't think there is any stretch of imagination by which this could be organised.
In principle it is possible for some part of a system to reflect in detail the whole system, as happens in fractals, but that happens because the system was from the outset designed to have that property. Actually what you want is that the encoding is at some higher level of abstraction (your knowledge about the universe should not just be a scaled copy of the universe), somewhat like a Gödel sentence is representing something about itself (namely the existence of a proof for itself). That example shows that again it is not something that fundamentally cannot exist at all, but the property of the Gödel sentence is something that happens only because it was designed that way from outside the system; within the system there is nothing to indicate that the Gödel sentence is actually talking about itself.
Also from an information-theoretic perspective, it is hard to see how one could make part of a system encode the whole system, or more specifically evolve from a state in which it does not to one in which it does.
By the way, this also addresses the question of how to reconcile a deterministic universe (or one with stochastic elements; it makes no fundamental difference) with the subjective notion of free will. Even if we are aware that our bodies, and therefore our thoughts and actions, are ultimately governed by the laws of nature and our initial state, over which we have no control at all, there is no way in which subjectively that determinism influences the options when we are deciding what to do, because it is impossible to have that higher level information about our state encoded inside our own brain. A chess computer may be programmed to ultimately decide according to a fixed algorithm to choose what it deems to be the best move (it cannot do anything else), that fact is in no way constraining the set of options that it must explore and weigh in order to find that move.
A: 
So assuming we know all the laws of physics in differential equation form, and I know the current state of the universe completely...

Pheew... what an optimistic thought! Without going into the general law of physics with the problem of consistency between relativity theory and quantum mechanics, I shall just look at something I know better: meteorological forecast. Here most of the laws are well known: you have a rather simple fluid mechanic question, a bit of water thermodynamics, and solar radiations for an even smaller part. The parameters to handle are rather simple: temperature, pression, moisture, and wind speed. But as Navier-Stokes equations are unstable, a minimal error on the initial conditions leads to a plain wrong solution after no more than 15 days!
Said differently, it is currently impossible to fully describe the troposphere (first 15km of the atmosphere), and you assume that you know the current state of the universe!
My answer is simply that if you can know the current state of the universe completely you are with no doubt an almighty god, so you should be able to know what has happened since the big bang  and what will happen till the end of the universe...
Not speaking of the fact that what we know of physical laws already contains singularities where the currently known laws no longer apply: the black holes and the big bang itself...
On a logical and philosophical point of view, your question starts with an impossible assumption. From that on, the logic rules say that anything is possible: false implies anything is always true...
A: Let's not even talk about big bangs yet. Consider a simple non-linear ODE $\frac{dx}{dt}=-x^2$ with the condition $x(1)=1$. There is a unique maximal solution defined on a connected interval, which in this case is easily seen to be  $x(t)=\frac{1}{t}$ for $t\in (0,\infty)$. Ouch. Even for such a simple looking ODE, a simple non-linearity already implies that our solution blows up in a finite amount of time, and we can't continue 'backwards' beyond $t=0$. You as an observer living in the 'future', i.e living in $(0,\infty)$ can no longer ask "what happened at $t=-1$?" The answer is that you can't say anything. Note that you can also cook up examples of ODEs for which solutions only exist for a finite interval of time $(t_1,t_2)$, and blowup as $t\to t_2^-$ or as $t\to t_1^+$.
The Einstein equations (which are PDEs, not merely ODEs) are a much bigger nonlinear mess. It is actually a general feature of nonlinear equations that solutions usually blow up in a finite amount of time. Of course, certain nonlinear equations have global-in-time existence of solutions, but a-priori, there's no reason you should expect them to have that nice property. For instance, in the FRW solution of Einstein's equations, the scale factor $a(t)$ vanishes as $t\to t_0$ (if you plug in some simple matter models you can even see this analytically), and doing a bunch more calculations, you can show this implies somme of the curvature components blow up. What this says is the Lorentzian metric cannot be extended in a $C^2$ sense. We can try to refine our notion of solution and singularity, but that would require a deep dive into the harshness of Sobolev spaces etc, and I don't want to open that can of worms here or now.
Anyway, my simple point is that it is very common to have ODEs which only have solutions that exist for a finite amount of time, so your central claim of

It seems to me that I could plug these into my differential equations and find out the state of the universe infinitely far back or infinitely in the future.

is just not true.

Edit:
@jensenpaull good point, and I was debating whether or not I should have elaborated on it originally, but since you asked, I’ll do so now. Are there functions that satisfy the ODE $\frac{dx}{dt}=-x^2$ which are defined on a larger domain? Absolutely! The general solution is $x(t)=\frac{1}{t}+C(t)$, where $C(t)$ is constant on $(0,\infty)$, and a perhaps different constant on $(-\infty,0)$. So, we we have completely lost uniqueness. But, why is this physically (and even mathematically in some regards) such a big deal?
In Physics, we do experiments, and that means we have only access to things ‘here and now’ (let’s gloss over technical (but fundamental) issues and say we have the ability to gather perfect experimental data). One of the goals of Physics is to use this information, and predict what happens in the future/past. But if we lose uniqueness, then it means our perfect initial conditions are still insufficient to nail down what exactly happened/will happen, which is a sign that we don’t know everything. We are talking about dynamics here, so our perfect knowledge ‘initially’ should be all that we require to talk about existence and uniqueness of solutions (Otherwise, our theory is not well-posed). So, anything which is not uniquely predicted by our initial conditions cannot in any sense be considered physically relevant. Btw, such ‘well-posedness’ (in a certain class) questions are taken for granted in Physics, and occupy Mathematicians (heck the Navier-Stokes Millenium problem is roughly speaking a question of well-posedness in a smooth setting). Dynamics is everywhere:

*

*Newton’s laws are 2nd order ODEs and require require two initial conditions (position, velocity). From there, we turn on our ODE solver, and see what the result is.

*Maxwell’s electrodynamics: although in elementary E&M we simply solve various equations using symmetry, the fundamental idea is these are (linear, coupled) evolution equations for a pair of vector fields, which means we prescribe certain initial conditions (and boundary conditions) and then solve.

*GR: initially, there was lots of confusion regarding what exactly a solution is. It wasn’t until the work of Choquet Bruhat (and Geroch) that we finally understood the dynamical formulation of Einstein’s equations, and that we had a good well-posedness statement and a firm understanding of how the initial conditions (a 3-manifold, a Riemannian metric, and a symmetric $(0,2)$-tensor field which is the to-be second-fundamental form of the embedding) give rise to a unique maximal solution (which is globally hyperbolic).

So, my first reason for why we don’t continue past $t=0$ (though of course, the reasoning is not really specific to that ODE alone) has been that dynamics should be uniquely predicted by initial conditions. Hence, it makes no physical sense to go beyond $t=0$. The second reason is that in physics, nothing is ‘truly infinite’, and if it is, then our interpretation is that we don’t yet have a complete understanding of what’s going on. So, rather than trying to fix our solution, we should fix our equations (e.g maybe the ODE isn’t very physical). But before we throw out our equations, we may wonder: have we been too restrictive in our notion of solution? For instance, maybe it is too much to require solutions to be $C^1$. Could we for instance require only weaker regularity of $L^2=H^0$ or $H^1$? Well, $H^1$-regularity is indeed more natural for many Physical purposes (because $H^1$-regularity means ‘energy stays finite’). However, for this solution, we can see that $\frac{1}{2}\int_0^{\infty}|x(t)|^2+|\dot{x}(t)|^2\,dt=\infty$. In fact, this is so bad that for any $\epsilon>0$, $\int_0^{\epsilon}[\dots]\,dt=\infty$, so the origin is a truly singular point that even energy blows up. So, there’s no physical sense in continuing past that point.
