Data Imputation in R with NAs in only one variable (categorical) I have data frame with 44,353 entries with 17 variables (4 categorical + 13 continuous). Out of all variables only 1 categorical variable (with 52 factors) has NAs

No of factors in the categorical variables are 1601, 6, 52 and 15

When I use missforest package it throws error that it cannot handle categorical predictors with more that 53 categories.
Please suggest an imputation method in R for best accuracy. Also since the variable to be imputed is categorical I would prefer to avoid methods that use regression techniques to impute values.
 A: You could use random hot deck imputation. Roughly, this is a method where missing values are replaced with values from an observation with "similar" values in the non-missing variables. For each missing value, the algorithm generates a pool of similar observations ("donors") and randomly chooses from them. The bigger the pools of donors and the bigger the number of datasets being imputed, the better. 
For example we might have data as follows
 var1  var2  var3  var4
1 B     Z     U     5.1
2 <NA>  Z     U     5.0
3 B     Z     U     4.9
4 A     W     U     5.2
5 B     W     U     4.8
6 C     U     T     6.2
7 B     C     T     5.2
8 B     T     T     6.1
9 B     S     T     6.0

Here we have one observation with a missing value in var1. The algorithm would identify observations 1 and 3 as donors since they both have the same values for the two other categorical variables and similar values for the numeric variable. Hence, B would be chosen as the imputed values. If instead row 1 was
1 A     Z     U     5.1

..then the algorithm would randomly choose either A or B as the imputed value.
The hot.deck package for R should be able to handle your situation. The only problem I see is that, since you only have ~45000 observations, it is clear that there are many combinations of the categorical variable levels that do not occur (since, if every combination occured, there would have to be a minimum of $1601 \times 6  \times 52  \times 15 = 7,492,680$ observations. So, for the observations with missing data, if the combinations of the categorical variables are not repeated at least $M$ times elsewhere in the data, and you are imputing $m$ datasets, if $m > M$ there will be insufficient donors to draw imputations from. If this occurs then the software will generate warnings. 
A: Do you need to impute NA's?
First I would ask if you really need to impute the missing values? If you intend to use the imputed set to train another model you might as well just add NA as a level. In my experience this is really the simplest solution when you have NA's in a categorical variable. Especially when NA's actually do mean something, which is quite common. But even if it does not it is easy, especially for random forests, to ignore that level if it is not predictive. 
This will add NA as a level in the factor.
dataset$varWithNAs <- addNA(dataset$varWithNAs)

Dummy encoding large categorical features
Regarding the problem with too many levels it seems to be the factor w 1601 levels that is your main problem. This is really a lot of levels and it is hard to give you any direct usage tips as little is stated about the variable. What you always can do in the case of too many levels is to transform the variable into many boolean (true, false) variables.
I'll give you an example.
dataset <- data.frame(x1 = sample(c('a','b','c'), 10, replace=T))
#     x1
# 1   c
# 2   b
# 3   a
# 4   a
# 5   b
# 6   c
# 7   a
# 8   a
# 9   b
# 10  c

You could use the caret package to create dummy variables for your factor levels. 
library(caret)
dummyObj <- dummyVars(~x1, dataset)
dummyset <- predict(dummyObj, dataset)
     x1.a x1.b x1.c
# 1     0    0    1
# 2     0    1    0
# 3     1    0    0
# 4     1    0    0
# 5     0    1    0
# 6     0    0    1
# 7     1    0    0
# 8     1    0    0
# 9     0    1    0
# 10    0    0    1

In your case it will make your feature vector quite a lot wider but it is actually what is done internally in a lot of, especially linear, models before training (although not in RF which is why you get this problem). If you look at eg. the glm package it transforms the dataset into dummy variables using the model.matrix function which does the same but adds an intercept term. Removing this intercept term will give you the same answer. And as model.matrix exists in the stats package you don't need to install anything.
model.matrix(~ x1 - 1, dataset) # -1 removes the intercept
#    x1a x1b x1c
# 1    0   0   1
# 2    0   1   0
# 3    1   0   0
# 4    1   0   0
# 5    0   1   0
# 6    0   0   1
# 7    1   0   0
# 8    1   0   0
# 9    0   1   0
# 10   0   0   1

If you find that your dataset get too many features now you should resort to the options Michael M gave in his answer to reduce the feature space. Chances are you have levels that never occur or several that are very similar in meaning and can be combined etc. Of course it is tedious to do this manually when you have so many levels. 
A: The magic word is feature construction. Statistics and machine learning almost always take some effort to prepare data before running an algorithm. Some "tricks" with respect to your particular problem:


*

*try to reduce the number of factor levels without losing too much information (e.g. combine rare levels)

*Convert ordered input factors to numeric (tree-based methods work much faster then).

*A possibility is also to manually dummy code an input factor which greatly reduces computational effort.


Some further hints:


*

*If the response variable is a factor, then a random forest does classification, not regression. 

*You can try out faster algoritms like multinomial logistic regression to fill in missings.

