I don’t really get what you mean with AGWN, is this simply that $n(t_i)$ are independent with $n(t_i) \sim N(0,\sigma^2)$?

The least squares estimator (which, as usual with a normal model, coincide with maximum likelihood estimator, see eg [this answer](https://stats.stackexchange.com/a/12565/8076)) is easy to find with numerical methods, here is a piece of R code:

    a <- 2; b <- 0.2; c <- -1
    t <- seq(-5,10, length=100)
    y <- a*exp(-b*t)+c+rnorm(length(t), sd=2)
    f <- function(par, t, y) { 
      a <- par[1]; b <- par[2]; c <- par[3]; 
      return(sum((a*exp(-b*t)+c-y)**2 ) ); }
    nlm( function(par) f(par, t, y), c(0,0,0))

The result of the last call is 

    $minimum
    [1] 430.8242

    $estimate
    [1]  2.0875336  0.1961210 -0.8672079

    $gradient
    [1]  1.301591e-05 -2.745537e-05  2.603429e-05

    $code
    [1] 1

    $iterations
    [1] 19

Our initial values (2, 0.2, -1) are estimated by (2.088, 0.196, -0.867). You can get a plot of the data points, the "true model" (red line) and the estimated model (dotted red line) as follows:

    plot(t,y)
    lines( t, a*exp(-b*t)+c, col="red")
    nlm( function(par) f(par, t, y), c(0,0,0))$estimate -> r
    lines( t, r[1]*exp(-r[2]*t)+r[3], col="red", lty=2)

![Data points, with in red line the true model, in dotted red line the fitted model][1]

Also, be sure not to miss [this related question](https://stats.stackexchange.com/questions/6720/estimation-of-exponential-model) already quoted by whuber in the comments.

**A few hints for numerical optimization**

In the above code the numerical optimization is done using `nlm`, a function of R. Here are a few hints for a more elementary solution.

If $b\ne 0$ is fixed, it is easy to find $a$ and $c$ minimizing $f(a,b,c) = \sum_{i=1}^n (a \exp(-b\cdot t_i) +c -y_i)^2$: this a the classical least squares for the linear model.  Namely, letting $x_i = \exp(-b\cdot t_i)$, we find
$$ \begin{array}{rcccl}
a &=& a(b) &=& { n \sum_i y_i x_i - \left( \sum_i y_i \right)\left(\sum_i x_i  \right)\over n \sum x_i^2 - \left(\sum_i x_i \right)^2 }, \\
c &=& c(b) &=& { \sum_i y_i \sum_i x_i^2 - \left( \sum_i y_i x_i \right)\left(\sum_i x_i  \right)\over n \sum x_i^2 - \left(\sum_i x_i \right)^2 }. \\ 
\end{array}$$

Then set 
$$ g(b) = f(a(b), b, c(b)).$$
The optimization problem is now reduced to find the minimum of $g(b)$. This can be done either using [Newton method](http://en.wikipedia.org/wiki/Newton%27s_method_in_optimization) or a [ternary search](http://en.wikipedia.org/wiki/Ternary_search).

  [1]: https://i.sstatic.net/L1Uui.png