miktex-makepk: PK font Alegreya-Regular-lf-t1--base could not be created

Dvips.map and friends do not know about Alegreya, so instead of loading type 1 fonts, MiKTeX tries to make bitmaps and doesn't know how to make them.

Check if you have a text file named updmap.cfg in C:\Users\Your_name\AppData\Roaming\MiKTeX\2.9\miktex\config. If not, create one. Add the following lines in that file:

Map Alegreya.map

Then execute as administrator this command line:

updmap.exe -verbose

The -verbose switch is there to check what's happening. That's all. You can check everything's OK by looking at the contents of psfonts.map, in C:\Users\Your_name\AppData\Local\MiKTeX\2.9\dvips\config. It should contain references to Alegreya.

Alternatively, you can compile with XeLateX or LuaLaTeX: you have to remove inputenc loading, save you document in UTF-8 format (no \usepackage[utf8]{inputenc}!) and write these lines in your preamble

\usepackage{fontspec}  
\setmainfont{Alegreya}

Addendum: there seems to be a typo in line 233 of (MiKTeX's version of) alegreya.sty, which begins with:

\def*\useosf{...

and should begin with

\def\useosf{...

I had the same problem but with the Classico font. My error message was

FATAL miktex-makemf - The URWClassico-Bold-lf-t1--base source file could not be found

followed by

FATAL miktex-makepk - PK font URWClassico-Bold-lf-t1--base could not be created.

I solved it by following the instructions at https://www.tug.org/fonts/fontinstall.html under "4. Font map files: telling TeX about the new font"

In short, run initexmf --edit-config-file updmap, add Map <mapfile> (in my case Map Classico.map) to the file, run initexmf --mkmaps and finally initexmf --update-fndb.