Java 8 Date equivalent to Joda's DateTimeFormatterBuilder with multiple parser formats?

There is no direct facility to do this, but you can use optional sections. Optional sections are enclosed inside squared brackets []. This allows for the whole section of the String to parse to be missing.

DateTimeFormatter formatter = DateTimeFormatter.ofPattern(""
    + "[yyyy/MM/dd HH:mm:ss.SSSSSS]"
    + "[yyyy-MM-dd HH:mm:ss[.SSS]]"
    + "[ddMMMyyyy:HH:mm:ss.SSS[ Z]]"
);

This formatter defines 3 grand optional sections for the three main patterns you have. Each of them is inside its own optional section.

Working demo code:

public static void main(String[] args) {
    DateTimeFormatter formatter = DateTimeFormatter.ofPattern(""
        + "[yyyy/MM/dd HH:mm:ss.SSSSSS]"
        + "[yyyy-MM-dd HH:mm:ss[.SSS]]"
        + "[ddMMMyyyy:HH:mm:ss.SSS[ Z]]"
    , Locale.ENGLISH);
    System.out.println(LocalDateTime.parse("2016/03/23 22:00:00.256145", formatter));
    System.out.println(LocalDateTime.parse("2016-03-23 22:00:00", formatter));
    System.out.println(LocalDateTime.parse("2016-03-23 22:00:00.123", formatter));
    System.out.println(LocalDateTime.parse("23Mar2016:22:00:00.123", formatter));
    System.out.println(LocalDateTime.parse("23Mar2016:22:00:00.123 -0800", formatter));
}

As an alternative answer to Tunaki, you can also use DateTimeFormatterBuilder:

DateTimeFormatter dateFormatter = new DateTimeFormatterBuilder()
  .appendPattern("[yyyy]")
  .appendPattern("[M/d/yyyy]")
  .parseDefaulting(ChronoField.MONTH_OF_YEAR, 1)
  .parseDefaulting(ChronoField.DAY_OF_MONTH, 1)
  .toFormatter()

Iterating over @Tunaki's solution, using streams, when the code need to accept different patterns in a configurable way :

DateTimeFormatter dateTimeFormatter = dateFormats.stream()
        .map(DateTimeFormatter::ofPattern)
        .reduce(new DateTimeFormatterBuilder(), 
                DateTimeFormatterBuilder::appendOptional, 
                (f1, f2) -> f1.append(f2.toFormatter()))
        .toFormatter();

In this case I don't care about the combiner part of the reducer, but I need it in the signature so I made the combiner correct.

This code would virtually equivalent to if the above patterns (yyyy/MM/dd HH:mm:ss.SSSSSS, yyyy-MM-dd HH:mm:ss[.SSS], ddMMMyyyy:HH:mm:ss.SSS[ Z]) would be fed to the stream :

DateTimeFormatter formatter = new DateTimeFormatterBuilder()
    .appendOptional(DateTimeFormatter.ofPattern("yyyy/MM/dd HH:mm:ss.SSSSSS")
    .appendOptional(DateTimeFormatter.ofPattern("yyyy-MM-dd HH:mm:ss[.SSS]"
    .appendOptional(DateTimeFormatter.ofPattern("ddMMMyyyy:HH:mm:ss.SSS[ Z]")
    .toFormatter();