I am working with R version 3.6.3, Bioconductor version 3.10.

When I try to install the "TIN" package, the following appears:

> install("TIN")
Bioconductor version 3.10
  (BiocManager 1.30.10), R 3.6.3
  (2020-02-29)
Installing package(s) 'TIN'
trying URL 'https://bioconductor.org/packages/3.10/bioc/bin/windows/contrib/3.6/TIN_1.18.0.zip'
Content type 'application/zip' length 4812027 bytes (4.6 MB)
downloaded 4.6 MB

package ‘TIN’ successfully unpacked and MD5 sums checked

The downloaded binary packages are in
    C:\Users\Antonia\AppData\Local\Temp\RtmpcLN0kN\downloaded_packages
Old packages: 'ellipsis', 'glue',
  'RcppArmadillo', 'vctrs',
  'lattice', 'MASS', 'nlme',
  'survival'
Update all/some/none? [a/s/n]: 
a

  There are binary versions
  available but the source
  versions are later:
                   binary      source
ellipsis            0.3.0       0.3.1
RcppArmadillo 0.9.870.2.0 0.9.880.1.0
              needs_compilation
ellipsis                   TRUE
RcppArmadillo              TRUE

  Binaries will be installed
trying URL 'https://cran.rstudio.com/bin/windows/contrib/3.6/ellipsis_0.3.0.zip'
Content type 'application/zip' length 44547 bytes (43 KB)
downloaded 43 KB

trying URL 'https://cran.rstudio.com/bin/windows/contrib/3.6/glue_1.4.1.zip'
Content type 'application/zip' length 153729 bytes (150 KB)
downloaded 150 KB

trying URL 'https://cran.rstudio.com/bin/windows/contrib/3.6/RcppArmadillo_0.9.870.2.0.zip'
Content type 'application/zip' length 2371431 bytes (2.3 MB)
downloaded 2.3 MB

trying URL 'https://cran.rstudio.com/bin/windows/contrib/3.6/vctrs_0.3.0.zip'
Content type 'application/zip' length 1123207 bytes (1.1 MB)
downloaded 1.1 MB

package ‘ellipsis’ successfully unpacked and MD5 sums checked
Warning: cannot remove prior installation of package ‘ellipsis’
Warning: restored ‘ellipsis’
package ‘glue’ successfully unpacked and MD5 sums checked
Warning: cannot remove prior installation of package ‘glue’
Warning: restored ‘glue’
package ‘RcppArmadillo’ successfully unpacked and MD5 sums checked
package ‘vctrs’ successfully unpacked and MD5 sums checked
Warning: cannot remove prior installation of package ‘vctrs’
Warning: restored ‘vctrs’

The downloaded binary packages are in
    C:\Users\*******\AppData\Local\Temp\RtmpcLN0kN\downloaded_packages
trying URL 'https://cran.rstudio.com/bin/windows/contrib/3.6/lattice_0.20-41.zip'
Content type 'application/zip' length 1174629 bytes (1.1 MB)
downloaded 1.1 MB

trying URL 'https://cran.rstudio.com/bin/windows/contrib/3.6/MASS_7.3-51.6.zip'
Content type 'application/zip' length 1190824 bytes (1.1 MB)
downloaded 1.1 MB

trying URL 'https://cran.rstudio.com/bin/windows/contrib/3.6/nlme_3.1-147.zip'
Content type 'application/zip' length 2370362 bytes (2.3 MB)
downloaded 2.3 MB

trying URL 'https://cran.rstudio.com/bin/windows/contrib/3.6/survival_3.1-12.zip'
Content type 'application/zip' length 7783773 bytes (7.4 MB)
downloaded 7.4 MB

package ‘lattice’ successfully unpacked and MD5 sums checked
Warning: cannot remove prior installation of package ‘lattice’
Warning: restored ‘lattice’
package ‘MASS’ successfully unpacked and MD5 sums checked
package ‘nlme’ successfully unpacked and MD5 sums checked
Warning: cannot remove prior installation of package ‘nlme’
Warning: restored ‘nlme’
package ‘survival’ successfully unpacked and MD5 sums checked
Warning: cannot remove prior installation of package ‘survival’
Warning: restored ‘survival’

The downloaded binary packages are in
    C:\Users\*******\AppData\Local\Temp\RtmpcLN0kN\downloaded_packages
Warning messages:
1: In file.copy(savedcopy, lib, recursive = TRUE) :
  problem copying C:\Users\*******\Documents\R\win-library\3.6\00LOCK\ellipsis\libs\x64\ellipsis.dll to C:\Users\*******\Documents\R\win-library\3.6\ellipsis\libs\x64\ellipsis.dll: Permission denied
2: In file.copy(savedcopy, lib, recursive = TRUE) :
  problem copying C:\Users\*******a\Documents\R\win-library\3.6\00LOCK\glue\libs\x64\glue.dll to C:\Users\*******\Documents\R\win-library\3.6\glue\libs\x64\glue.dll: Permission denied
3: In file.copy(savedcopy, lib, recursive = TRUE) :
  problem copying C:\Users\*******\Documents\R\win-library\3.6\00LOCK\vctrs\libs\x64\vctrs.dll to C:\Users\*******\Documents\R\win-library\3.6\vctrs\libs\x64\vctrs.dll: Permission denied
4: In file.copy(savedcopy, lib, recursive = TRUE) :
  problem copying C:\Program Files\R\R-3.6.3\library\00LOCK\lattice\libs\x64\lattice.dll to C:\Program Files\R\R-3.6.3\library\lattice\libs\x64\lattice.dll: Permission denied
5: In file.copy(savedcopy, lib, recursive = TRUE) :
  problem copying C:\Program Files\R\R-3.6.3\library\00LOCK\nlme\libs\x64\nlme.dll to C:\Program Files\R\R-3.6.3\library\nlme\libs\x64\nlme.dll: Permission denied
6: In file.copy(savedcopy, lib, recursive = TRUE) :
  problem copying C:\Program Files\R\R-3.6.3\library\00LOCK\survival\libs\x64\survival.dll to C:\Program Files\R\R-3.6.3\library\survival\libs\x64\survival.dll: Permission denied

The package appears in my directory, but will not let me attach it to the workspace:

> library(TIN)
Error: package or namespace load failed for ‘TIN’ in loadNamespace(i, c(lib.loc, .libPaths()), versionCheck = vI[[i]]):
 there is no package called ‘GO.db’

I have run the script as admin and I have not had any trouble with other packages, which all install and attach fine. When I have searched for the GO.db package in my documents, it is found in the same file path as the TIN package (I'm not sure if this matters). I would really appreciate any feedback.

Does library(GO.db) work? If not, does BiocManager::install("GO.db") work? If not, report how it fails.

You should not install packages as an Administrator. I suggest cleaning up your libraries so that packages are only installed as a regular user.