Convert timeseries stack of GTiff raster to single NetCDF

Here's some python code that does what you want, reading GDAL files that represent data at specific times and writing to a single NetCDF file that is CF-Compliant

#!/usr/bin/env python
'''
Convert a bunch of GDAL readable grids to a NetCDF Time Series.
Here we read a bunch of files that have names like:
/usgs/data0/prism/1890-1899/us_tmin_1895.01
/usgs/data0/prism/1890-1899/us_tmin_1895.02
...
/usgs/data0/prism/1890-1899/us_tmin_1895.12
'''

import numpy as np
import datetime as dt
import os
import gdal
import netCDF4
import re

ds = gdal.Open('/usgs/data0/prism/1890-1899/us_tmin_1895.01')
a = ds.ReadAsArray()
nlat,nlon = np.shape(a)

b = ds.GetGeoTransform() #bbox, interval
lon = np.arange(nlon)*b[1]+b[0]
lat = np.arange(nlat)*b[5]+b[3]


basedate = dt.datetime(1858,11,17,0,0,0)

# create NetCDF file
nco = netCDF4.Dataset('time_series.nc','w',clobber=True)

# chunking is optional, but can improve access a lot: 
# (see: http://www.unidata.ucar.edu/blogs/developer/entry/chunking_data_choosing_shapes)
chunk_lon=16
chunk_lat=16
chunk_time=12

# create dimensions, variables and attributes:
nco.createDimension('lon',nlon)
nco.createDimension('lat',nlat)
nco.createDimension('time',None)
timeo = nco.createVariable('time','f4',('time'))
timeo.units = 'days since 1858-11-17 00:00:00'
timeo.standard_name = 'time'

lono = nco.createVariable('lon','f4',('lon'))
lono.units = 'degrees_east'
lono.standard_name = 'longitude'

lato = nco.createVariable('lat','f4',('lat'))
lato.units = 'degrees_north'
lato.standard_name = 'latitude'

# create container variable for CRS: lon/lat WGS84 datum
crso = nco.createVariable('crs','i4')
csro.long_name = 'Lon/Lat Coords in WGS84'
crso.grid_mapping_name='latitude_longitude'
crso.longitude_of_prime_meridian = 0.0
crso.semi_major_axis = 6378137.0
crso.inverse_flattening = 298.257223563

# create short integer variable for temperature data, with chunking
tmno = nco.createVariable('tmn', 'i2',  ('time', 'lat', 'lon'), 
   zlib=True,chunksizes=[chunk_time,chunk_lat,chunk_lon],fill_value=-9999)
tmno.units = 'degC'
tmno.scale_factor = 0.01
tmno.add_offset = 0.00
tmno.long_name = 'minimum monthly temperature'
tmno.standard_name = 'air_temperature'
tmno.grid_mapping = 'crs'
tmno.set_auto_maskandscale(False)

nco.Conventions='CF-1.6'

#write lon,lat
lono[:]=lon
lato[:]=lat

pat = re.compile('us_tmin_[0-9]{4}\.[0-9]{2}')
itime=0

#step through data, writing time and data to NetCDF
for root, dirs, files in os.walk('/usgs/data0/prism/1890-1899/'):
    dirs.sort()
    files.sort()
    for f in files:
        if re.match(pat,f):
            # read the time values by parsing the filename
            year=int(f[8:12])
            mon=int(f[13:15])
            date=dt.datetime(year,mon,1,0,0,0)
            print(date)
            dtime=(date-basedate).total_seconds()/86400.
            timeo[itime]=dtime
           # min temp
            tmn_path = os.path.join(root,f)
            print(tmn_path)
            tmn=gdal.Open(tmn_path)
            a=tmn.ReadAsArray()  #data
            tmno[itime,:,:]=a
            itime=itime+1

nco.close()

GDAL and NetCDF4 Python can be a bit of a pain to build, but the good news is that they are part of most scientific python distributions (Python(x,y), Enthought Python Distribution, Anaconda, ...)

Update: I haven't done polar stereographic in CF-compliant NetCDF yet, but I should look something like this. Here I've assumed that central_meridian and latitude_of_origin in GDAL are the same as straight_vertical_longitude_from_pole and latitude_of_projection_origin in CF:

#!/usr/bin/env python
'''
Convert a bunch of GDAL readable grids to a NetCDF Time Series.
Here we read a bunch of files that have names like:
/usgs/data0/prism/1890-1899/us_tmin_1895.01
/usgs/data0/prism/1890-1899/us_tmin_1895.02
...
/usgs/data0/prism/1890-1899/us_tmin_1895.12
'''

import numpy as np
import datetime as dt
import os
import gdal
import netCDF4
import re

ds = gdal.Open('/usgs/data0/prism/1890-1899/us_tmin_1895.01')
a = ds.ReadAsArray()
ny,nx = np.shape(a)

b = ds.GetGeoTransform() #bbox, interval
x = np.arange(nx)*b[1]+b[0]
y = np.arange(ny)*b[5]+b[3]


basedate = dt.datetime(1858,11,17,0,0,0)

# create NetCDF file
nco = netCDF4.Dataset('time_series.nc','w',clobber=True)

# chunking is optional, but can improve access a lot: 
# (see: http://www.unidata.ucar.edu/blogs/developer/entry/chunking_data_choosing_shapes)
chunk_x=16
chunk_y=16
chunk_time=12

# create dimensions, variables and attributes:
nco.createDimension('x',nx)
nco.createDimension('y',ny)
nco.createDimension('time',None)
timeo = nco.createVariable('time','f4',('time'))
timeo.units = 'days since 1858-11-17 00:00:00'
timeo.standard_name = 'time'

xo = nco.createVariable('x','f4',('x'))
xo.units = 'm'
xo.standard_name = 'projection_x_coordinate'

yo = nco.createVariable('y','f4',('y'))
yo.units = 'm'
yo.standard_name = 'projection_y_coordinate'

# create container variable for CRS: x/y WGS84 datum
crso = nco.createVariable('crs','i4')
crso.grid_mapping_name='polar_stereographic'
crso.straight_vertical_longitude_from_pole = -45.
crso.latitude_of_projection_origin = 70.
crso.scale_factor_at_projection_origin = 1.0
crso.false_easting = 0.0
crso.false_northing = 0.0
crso.semi_major_axis = 6378137.0
crso.inverse_flattening = 298.257223563

# create short integer variable for temperature data, with chunking
tmno = nco.createVariable('tmn', 'i2',  ('time', 'y', 'x'), 
   zlib=True,chunksizes=[chunk_time,chunk_y,chunk_x],fill_value=-9999)
tmno.units = 'degC'
tmno.scale_factor = 0.01
tmno.add_offset = 0.00
tmno.long_name = 'minimum monthly temperature'
tmno.standard_name = 'air_temperature'
tmno.grid_mapping = 'crs'
tmno.set_auto_maskandscale(False)

nco.Conventions='CF-1.6'

#write x,y
xo[:]=x
yo[:]=y

pat = re.compile('us_tmin_[0-9]{4}\.[0-9]{2}')
itime=0

#step through data, writing time and data to NetCDF
for root, dirs, files in os.walk('/usgs/data0/prism/1890-1899/'):
    dirs.sort()
    files.sort()
    for f in files:
        if re.match(pat,f):
            # read the time values by parsing the filename
            year=int(f[8:12])
            mon=int(f[13:15])
            date=dt.datetime(year,mon,1,0,0,0)
            print(date)
            dtime=(date-basedate).total_seconds()/86400.
            timeo[itime]=dtime
           # min temp
            tmn_path = os.path.join(root,f)
            print(tmn_path)
            tmn=gdal.Open(tmn_path)
            a=tmn.ReadAsArray()  #data
            tmno[itime,:,:]=a
            itime=itime+1

nco.close()

It's easy to put them in a single NetCDF with GDAL utilities, example below. But you don't get the temporal dimension/other metadata of @RichSignell's answer. The tiffs just get dumped into subdatasets.

C:\remotesensing\testdata>dir /b ndvi*.tif
ndvi1.tif
ndvi2.tif
ndvi3.tif

C:\remotesensing\testdata>gdalbuildvrt -separate ndvi.vrt ndvi*.tif
0...10...20...30...40...50...60...70...80...90...100 - done.

C:\remotesensing\testdata>gdal_translate -of netcdf ndvi.vrt ndvi.nc
Input file size is 96, 88
0...10...20...30...40...50...60...70...80...90...100 - done.

C:\remotesensing\testdata>gdalinfo ndvi.nc
Driver: netCDF/Network Common Data Format
Files: ndvi.nc
Size is 512, 512
Coordinate System is `'
Metadata:
  NC_GLOBAL#Conventions=CF-1.5
  NC_GLOBAL#GDAL=GDAL 1.10.0, released 2013/04/24
  NC_GLOBAL#history=Wed Sep 04 09:49:11 2013: GDAL CreateCopy( ndvi.nc, ... )
Subdatasets:
  SUBDATASET_1_NAME=NETCDF:"ndvi.nc":Band1
  SUBDATASET_1_DESC=[88x96] Band1 (32-bit floating-point)
  SUBDATASET_2_NAME=NETCDF:"ndvi.nc":Band2
  SUBDATASET_2_DESC=[88x96] Band2 (32-bit floating-point)
  SUBDATASET_3_NAME=NETCDF:"ndvi.nc":Band3
  SUBDATASET_3_DESC=[88x96] Band3 (32-bit floating-point)
Corner Coordinates:
Upper Left  (    0.0,    0.0)
Lower Left  (    0.0,  512.0)
Upper Right (  512.0,    0.0)
Lower Right (  512.0,  512.0)
Center      (  256.0,  256.0)

C:\remotesensing\testdata>gdalinfo NETCDF:"ndvi.nc":Band1
Driver: netCDF/Network Common Data Format
Files: ndvi.nc
Size is 96, 88
Coordinate System is:
GEOGCS["GCS_GDA_1994",
    DATUM["Geocentric_Datum_of_Australia_1994",
        SPHEROID["GRS 1980",6378137,298.2572221010002,
            AUTHORITY["EPSG","7019"]],
        AUTHORITY["EPSG","6283"]],
    PRIMEM["Greenwich",0],
    UNIT["degree",0.0174532925199433]]
Origin = (115.810500000000000,-32.260249999999999)
Pixel Size = (0.000250000000000,-0.000250000000000)
Metadata:
  Band1#_FillValue=0
  Band1#grid_mapping=crs
  Band1#long_name=GDAL Band Number 1
  crs#GeoTransform=115.8105 0.00025 0 -32.26025 0 -0.00025
  crs#grid_mapping_name=latitude_longitude
  crs#inverse_flattening=298.2572221010002
  crs#longitude_of_prime_meridian=0
  crs#semi_major_axis=6378137
  crs#spatial_ref=GEOGCS["GCS_GDA_1994",DATUM["Geocentric_Datum_of_Australia_1994",SPHEROID["GRS 1980",6378137,298.2572221010002,AUTHORITY["EPSG","7019"]],AUTHORITY["EPSG","6283"]],PRIMEM["Greenwich",0],UNIT["degree",0.0174532925199433]]
  lat#long_name=latitude
  lat#standard_name=latitude
  lat#units=degrees_north
  lon#long_name=longitude
  lon#standard_name=longitude
  lon#units=degrees_east
  NC_GLOBAL#Conventions=CF-1.5
  NC_GLOBAL#GDAL=GDAL 1.10.0, released 2013/04/24
  NC_GLOBAL#history=Wed Sep 04 09:49:11 2013: GDAL CreateCopy( ndvi.nc, ... )
Corner Coordinates:
Upper Left  ( 115.8105000, -32.2602500) (115d48'37.80"E, 32d15'36.90"S)
Lower Left  ( 115.8105000, -32.2822500) (115d48'37.80"E, 32d16'56.10"S)
Upper Right ( 115.8345000, -32.2602500) (115d50' 4.20"E, 32d15'36.90"S)
Lower Right ( 115.8345000, -32.2822500) (115d50' 4.20"E, 32d16'56.10"S)
Center      ( 115.8225000, -32.2712500) (115d49'21.00"E, 32d16'16.50"S)
Band 1 Block=96x1 Type=Float32, ColorInterp=Undefined
  NoData Value=0
  Metadata:
    _FillValue=0
    grid_mapping=crs
    long_name=GDAL Band Number 1
    NETCDF_VARNAME=Band1