I tried this,
It is single threaded.
public static void main(String args[]) throws Exception {
    Object obj = new Object();
    try {
        synchronized (obj) {
            obj.wait();
            System.out.println("after wait()");
        }
    } catch (Exception ignored) {
    } finally {
        System.out.println("finally");
    }
}
The main Thread will be on wait state forever, hence finally will never be called,
so console output will not print String: after wait() or finally
Agreed with @Stephen C, the above example is one of the 3rd case mention here:
Adding some more such infinite loop possibilities in following code:
// import java.util.concurrent.Semaphore;
public static void main(String[] args) {
    try {
        // Thread.sleep(Long.MAX_VALUE);
        // Thread.currentThread().join();
        // new Semaphore(0).acquire();
        // while (true){}
        System.out.println("after sleep join semaphore exit infinite while loop");
    } catch (Exception ignored) {
    } finally {
        System.out.println("finally");
    }
}
Case 2: If the JVM crashes first
import sun.misc.Unsafe;
import java.lang.reflect.Field;
public static void main(String args[]) {
    try {
        unsafeMethod();
        //Runtime.getRuntime().halt(123);
        System.out.println("After Jvm Crash!");
    } catch (Exception e) {
    } finally {
        System.out.println("finally");
    }
}
private static void unsafeMethod() throws NoSuchFieldException, IllegalAccessException {
    Field f = Unsafe.class.getDeclaredField("theUnsafe");
    f.setAccessible(true);
    Unsafe unsafe = (Unsafe) f.get(null);
    unsafe.putAddress(0, 0);
}
Ref: How do you crash a JVM?
Case 6: If finally block is going to be executed by daemon Thread and all other non-daemon Threads exit before finally is called.
public static void main(String args[]) {
    Runnable runnable = new Runnable() {
        @Override
        public void run() {
            try {
                printThreads("Daemon Thread printing");
                // just to ensure this thread will live longer than main thread
                Thread.sleep(10000);
            } catch (Exception e) {
            } finally {
                System.out.println("finally");
            }
        }
    };
    Thread daemonThread = new Thread(runnable);
    daemonThread.setDaemon(Boolean.TRUE);
    daemonThread.setName("My Daemon Thread");
    daemonThread.start();
    printThreads("main Thread Printing");
}
private static synchronized void printThreads(String str) {
    System.out.println(str);
    int threadCount = 0;
    Set<Thread> threadSet = Thread.getAllStackTraces().keySet();
    for (Thread t : threadSet) {
        if (t.getThreadGroup() == Thread.currentThread().getThreadGroup()) {
            System.out.println("Thread :" + t + ":" + "state:" + t.getState());
            ++threadCount;
        }
    }
    System.out.println("Thread count started by Main thread:" + threadCount);
    System.out.println("-------------------------------------------------");
}
output: This does not print "finally" which implies "Finally block" in "daemon thread" did not execute  
main Thread Printing  
Thread :Thread[My Daemon Thread,5,main]:state:BLOCKED  
Thread :Thread[main,5,main]:state:RUNNABLE  
Thread :Thread[Monitor Ctrl-Break,5,main]:state:RUNNABLE   
Thread count started by Main thread:3  
-------------------------------------------------  
Daemon Thread printing  
Thread :Thread[My Daemon Thread,5,main]:state:RUNNABLE  
Thread :Thread[Monitor Ctrl-Break,5,main]:state:RUNNABLE  
Thread count started by Main thread:2  
-------------------------------------------------  
Process finished with exit code 0