You can achieve the desired behavior in BigQuery by using window functions along with some logic in CASE and LAST_VALUE. Below is a step-by-step solution to correctly assign the time_group based on your specified rules.

WITH grouped_data AS (
SELECT
    logged_time,
    -- Calculate the time difference in seconds from the previous row
    TIMESTAMP_DIFF(logged_time, LAG(logged_time) OVER (ORDER BY logged_time), SECOND) AS diff_seconds,
    -- Determine if a new group should start
    CASE 
      WHEN LAG(logged_time) OVER (ORDER BY logged_time) IS NULL OR 
           TIMESTAMP_DIFF(logged_time, LAG(logged_time) OVER (ORDER BY logged_time), SECOND) > 180 
      THEN logged_time
      ELSE NULL
    END AS new_group_marker
  FROM your_table_name
),
final_grouping AS (
  SELECT
    logged_time,
    -- Propagate the new group marker forward
    LAST_VALUE(new_group_marker IGNORE NULLS) OVER (
      ORDER BY logged_time ROWS BETWEEN UNBOUNDED PRECEDING AND CURRENT ROW
    ) AS time_group
  FROM grouped_data
)
SELECT
  logged_time,
  time_group
FROM final_grouping
ORDER BY logged_time;

Time difference:

LAG(logged_time) retrieves the logged_time of the previous row.
TIMESTAMP_DIFF calculates the difference in seconds between the current row and the previous one.

New group marker:

A CASE statement is used to define when a new group should start:
For the first row (LAG(logged_time) IS NULL).
When the time difference exceeds 3 minutes (180 seconds).

Group propagation:

LAST_VALUE with IGNORE NULLS propagates the most recent non-NULL new_group_marker forward, assigning it as the time_group for all rows within the same group.

Sorting and displaying results:

Finally, select and order the logged_time and time_group columns.

I hope this helps!